fix sync issue, frontend improvement

vue/src/components/__tests__/AppNav.spec.js

@@ -148,6 +148,26 @@ describe('AppNav', () => {
     confirmSpy.mockRestore()
   })
 
+  it('shows the unread count in the title when there are articles', async () => {
+    const { feeds } = useFeeds()
+    feeds.value = [
+      { id: 1, title: 'Article one', content: '', url: 'https://example.test/1', timestamp: '2026-01-01' },
+      { id: 2, title: 'Article two', content: '', url: 'https://example.test/2', timestamp: '2026-01-02' },
+    ]
+
+    const wrapper = mount(AppNav, { global: { plugins: [router] } })
+    await flushPromises()
+
+    expect(wrapper.find('.app-nav__title').text()).toContain('(2)')
+  })
+
+  it('hides the unread count when there are no articles', async () => {
+    const wrapper = mount(AppNav, { global: { plugins: [router] } })
+    await flushPromises()
+
+    expect(wrapper.find('.app-nav__unread').exists()).toBe(false)
+  })
+
   it('does not mark articles as read when the confirmation is dismissed', async () => {
     const { feeds } = useFeeds()
     feeds.value = [