diff --git a/jupyterlab/commands.py b/jupyterlab/commands.py
index b294f9fe5569..3819da821180 100644
--- a/jupyterlab/commands.py
+++ b/jupyterlab/commands.py
@@ -8,6 +8,7 @@
 import errno
 import glob
 import hashlib
+import itertools
 import json
 import logging
 import os
@@ -94,7 +95,6 @@ def wait(self):
         cache = []
         proc = self.proc
         kill_event = self._kill_event
-        import itertools
         spinner = itertools.cycle(['-', '\\', '|', '/'])
         while proc.poll() is None:
             sys.stdout.write(next(spinner))   # write the next character
@@ -1965,51 +1965,75 @@ def _compare_ranges(spec1, spec2, drop_prerelease1=False, drop_prerelease2=False
     if not r1.range or not r2.range:
         return
 
-    x1 = r1.set[0][0].semver
-    x2 = r1.set[0][-1].semver
-    y1 = r2.set[0][0].semver
-    y2 = r2.set[0][-1].semver
+    # Set return_value to a sentinel value
+    return_value = False
 
-    if x1.prerelease and drop_prerelease1:
-        x1 = x1.inc('patch')
+    # r1.set may be a list of ranges if the range involved an ||, so we need to test for overlaps between each pair.
+    for r1set, r2set in itertools.product(r1.set, r2.set):
+        x1 = r1set[0].semver
+        x2 = r1set[-1].semver
+        y1 = r2set[0].semver
+        y2 = r2set[-1].semver
 
-    if y1.prerelease and drop_prerelease2:
-        y1 = y1.inc('patch')
+        if x1.prerelease and drop_prerelease1:
+            x1 = x1.inc('patch')
 
-    o1 = r1.set[0][0].operator
-    o2 = r2.set[0][0].operator
+        if y1.prerelease and drop_prerelease2:
+            y1 = y1.inc('patch')
 
-    # We do not handle (<) specifiers.
-    if (o1.startswith('<') or o2.startswith('<')):
-        return
+        o1 = r1set[0].operator
+        o2 = r2set[0].operator
 
-    # Handle single value specifiers.
-    lx = lte if x1 == x2 else lt
-    ly = lte if y1 == y2 else lt
-    gx = gte if x1 == x2 else gt
-    gy = gte if x1 == x2 else gt
+        # We do not handle (<) specifiers.
+        if (o1.startswith('<') or o2.startswith('<')):
+            continue
 
-    # Handle unbounded (>) specifiers.
-    def noop(x, y, z):
-        return True
+        # Handle single value specifiers.
+        lx = lte if x1 == x2 else lt
+        ly = lte if y1 == y2 else lt
+        gx = gte if x1 == x2 else gt
+        gy = gte if x1 == x2 else gt
 
-    if x1 == x2 and o1.startswith('>'):
-        lx = noop
-    if y1 == y2 and o2.startswith('>'):
-        ly = noop
-
-    # Check for overlap.
-    if (gte(x1, y1, True) and ly(x1, y2, True) or
-        gy(x2, y1, True) and ly(x2, y2, True) or
-        gte(y1, x1, True) and lx(y1, x2, True) or
-        gx(y2, x1, True) and lx(y2, x2, True)
-       ):
-       return 0
-    if gte(y1, x2, True):
-        return 1
-    if gte(x1, y2, True):
-        return -1
-    raise AssertionError('Unexpected case comparing version ranges')
+        # Handle unbounded (>) specifiers.
+        def noop(x, y, z):
+            return True
+
+        if x1 == x2 and o1.startswith('>'):
+            lx = noop
+        if y1 == y2 and o2.startswith('>'):
+            ly = noop
+
+        # Check for overlap.
+        if (gte(x1, y1, True) and ly(x1, y2, True) or
+            gy(x2, y1, True) and ly(x2, y2, True) or
+            gte(y1, x1, True) and lx(y1, x2, True) or
+            gx(y2, x1, True) and lx(y2, x2, True)
+        ):
+            # if we ever find an overlap, we can return immediately
+            return 0
+
+        if gte(y1, x2, True):
+            if return_value is False:
+                # We can possibly return 1
+                return_value = 1
+            elif return_value == -1:
+                # conflicting information, so we must return None
+                return_value = None
+            continue
+
+        if gte(x1, y2, True):
+            if return_value is False:
+                return_value = -1
+            elif return_value == 1:
+                # conflicting information, so we must return None
+                return_value = None
+            continue
+
+        raise AssertionError('Unexpected case comparing version ranges')
+
+    if return_value is False:
+        return_value = None
+    return return_value
 
 
 def _is_disabled(name, disabled=[]):
@@ -2051,7 +2075,7 @@ def _format_compatibility_errors(name, version, errors):
 
 
 def _log_multiple_compat_errors(logger, errors_map):
-    """Log compatability errors for multiple extensions at once"""
+    """Log compatibility errors for multiple extensions at once"""
 
     outdated = []
     others = []
diff --git a/jupyterlab/tests/test_jupyterlab.py b/jupyterlab/tests/test_jupyterlab.py
index a4fd6002b35f..0c3cbafb76cb 100644
--- a/jupyterlab/tests/test_jupyterlab.py
+++ b/jupyterlab/tests/test_jupyterlab.py
@@ -28,7 +28,7 @@
     install_extension, uninstall_extension, list_extensions,
     build, link_package, unlink_package, build_check,
     disable_extension, enable_extension, get_app_info,
-    check_extension, _test_overlap, update_extension,
+    check_extension, _test_overlap, _compare_ranges, update_extension,
     AppOptions
 )
 from jupyterlab.coreconfig import CoreConfig, _get_default_core_data
@@ -613,6 +613,22 @@ def test_compatibility(self):
         assert _test_overlap('*', '0.6') is None
         assert _test_overlap('<0.6', '0.1') is None
 
+        assert _test_overlap('^1 || ^2', '^1')
+        assert _test_overlap('^1 || ^2', '^2')
+        assert _test_overlap('^1', '^1 || ^2')
+        assert _test_overlap('^2', '^1 || ^2')
+        assert _test_overlap('^1 || ^2', '^2 || ^3')
+        assert not _test_overlap('^1 || ^2', '^3 || ^4')
+        assert not _test_overlap('^2', '^1 || ^3')
+
+    def test_compare_ranges(self):
+        assert _compare_ranges('^1 || ^2', '^1') == 0
+        assert _compare_ranges('^1 || ^2', '^2 || ^3') == 0
+        assert _compare_ranges('^1 || ^2', '^3 || ^4') == 1
+        assert _compare_ranges('^3 || ^4', '^1 || ^2') == -1
+        assert _compare_ranges('^2 || ^3', '^1 || ^4') is None
+
+
     def test_install_compatible(self):
         core_data = _get_default_core_data()
         current_app_dep = core_data['dependencies']['@jupyterlab/application']