diff --git a/packages/downgrader/src/v3.1-to-v3.0.ts b/packages/downgrader/src/v3.1-to-v3.0.ts
index 2d99ccb..3d83809 100644
--- a/packages/downgrader/src/v3.1-to-v3.0.ts
+++ b/packages/downgrader/src/v3.1-to-v3.0.ts
@@ -1736,19 +1736,17 @@ function convertDocument(value: unknown, ctx: Context): unknown {
   return convertObject(value, ctx, DOCUMENT_FIELDS, finishDocument)
 }
 
-// A chain of bare `$ref`s from `schema` that comes back to it.
+// A chain of `$ref`s from `schema` that comes back to it. 3.0 has no `$ref`
+// with siblings, so each is bare.
 function leadsBack(schema: Record<string, unknown>, ctx: Context): boolean {
-  const seen = new Set<unknown>()
-  for (let next: unknown = schema; isRecord(next) && Object.keys(next).length === 1;) {
-    const ref = getRef(next)
-    if (ref === undefined || seen.has(next)) {
-      return false
-    }
-    seen.add(next)
-    next = resolvePointer(ref, ctx)
+  const seen = new Set<string>()
+  for (let ref = getRef(schema); ref !== undefined && !seen.has(ref);) {
+    seen.add(ref)
+    const next = resolvePointer(ref, ctx)
     if (next === schema) {
       return true
     }
+    ref = getRef(next)
   }
   return false
 }
diff --git a/packages/downgrader/tests/v3.1-to-v3.0.test.ts b/packages/downgrader/tests/v3.1-to-v3.0.test.ts
index 73889a3..87e086b 100644
--- a/packages/downgrader/tests/v3.1-to-v3.0.test.ts
+++ b/packages/downgrader/tests/v3.1-to-v3.0.test.ts
@@ -1403,6 +1403,17 @@ describe('downgradeSchemaV31ToV30', () => {
       expect(schemas({ A: { anyOf: [ref('A'), ref('A'), { type: 'null' }] } })).toEqual({
         A: { anyOf: [ref('A'), onlyNull] },
       })
+      // A cycle of objects in the target converts.
+      const cyclic: Record<string, any> = { anyOf: [{ type: 'null' }] }
+      cyclic.anyOf.unshift({ anyOf: [cyclic, { type: 'null' }] })
+      expect(schemas({ A: pair('B'), B: cyclic })?.A).toEqual(ref('B'))
+      // A `$ref` whose target moves in 3.0 leads nowhere.
+      expect(
+        schemas({
+          A: pair('B/properties/a'),
+          B: { type: ['object', 'string'], properties: { a: { type: 'null' } } },
+        })?.A,
+      ).toEqual(ref('B/properties/a'))
     })
 
     it('merges a branch in full where a cycle meets it before its conversion is done', () => {