diff --git a/packages/downgrader/src/v3.1-to-v3.0.ts b/packages/downgrader/src/v3.1-to-v3.0.ts
index f31cfb4..8c3b55c 100644
--- a/packages/downgrader/src/v3.1-to-v3.0.ts
+++ b/packages/downgrader/src/v3.1-to-v3.0.ts
@@ -513,18 +513,13 @@ function isUnevaluatedAdditional(schema: Record<string, unknown>): boolean {
 // property of its own, which 3.0 can express. A reusable enum is a `$ref` to it.
 function enumNames(schema: Record<string, unknown>, ctx: Context): string[] | undefined {
   // Followed alone or beside annotations, such as the `description` Zod writes
-  // for a described use, and along a chain of them, since other keywords
-  // beside a `$ref` narrow the names further.
-  let names = schema.propertyNames
-  const visited = new Set<unknown>()
-  for (
-    let ref = getRef(names);
-    ref !== undefined && isAnnotatedRef(names as Record<string, unknown>) && !visited.has(names);
-    ref = getRef(names)
-  ) {
-    visited.add(names)
-    names = resolvePointer(ref, ctx)
-  }
+  // for a described use, since other keywords beside it narrow the names
+  // further. A target that is a `$ref` again holds no names here.
+  const ref = getRef(schema.propertyNames)
+  const names =
+    ref !== undefined && isAnnotatedRef(schema.propertyNames as Record<string, unknown>)
+      ? resolvePointer(ref, ctx)
+      : schema.propertyNames
   if (!isRecord(names)) {
     return undefined
   }
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 3c00ab1..f212f11 100644
--- a/packages/downgrader/tests/v3.1-to-v3.0.test.ts
+++ b/packages/downgrader/tests/v3.1-to-v3.0.test.ts
@@ -2430,7 +2430,7 @@ describe('downgradeSchemaV31ToV30', () => {
     })
   })
 
-  it('follows a propertyNames $ref on through a $ref beside only annotations, and no further', () => {
+  it('follows a propertyNames $ref through one $ref only', () => {
     const r = '#/components/schemas/'
     const schemas = convert({
       components: {
@@ -2448,11 +2448,8 @@ describe('downgradeSchemaV31ToV30', () => {
         },
       },
     }).components?.schemas
-    expect(schemas?.ByAlias).toEqual({
-      type: 'object',
-      properties: { en: { type: 'string' }, fr: { type: 'string' } },
-      additionalProperties: false,
-    })
+    // A target that is a $ref again holds no names, so the record loses them.
+    expect(schemas?.ByAlias).toEqual({ type: 'object', additionalProperties: { type: 'string' } })
     expect(schemas?.NotEnglish).toEqual({})
   })