TF-4056 Persist filter search when change quick search input text

[dab246]

Issue

#4056

Resolved

• Demo

Screen.Recording.2025-09-24.at.4.33.19.PM.mov

• E2E Console:

🧪 Should see the same email when selecting filter and changing search input text
        ✅   1. waitUntilVisible widgets with type "TwakeWelcomeView".
        ✅   2. tap widgets with text "Use company server".
        ✅   3. waitUntilVisible widgets with type "LoginView".
        ✅   4. tap widgets with type "TextField" descending from widgets with key [<'dns_lookup_input_form'>].
        ✅   5. enterText widgets with type "TextField" descending from widgets with key [<'dns_lookup_input_form'>].
        ✅   6. waitUntilVisible widgets with text "bob".
        ✅   7. tap widgets with text "Next".
        ✅   8. waitUntilVisible widgets with key [<'base_url_form'>] descending from widgets with type "LoginView".
        ✅   9. enterText widgets with type "TextField" descending from widgets with key [<'base_url_form'>].
        ✅  10. waitUntilVisible widgets with text containing 1d37cdea321e.ngrok-free.app.
        ✅  11. tap widgets with text "Next".
        ✅  12. waitUntilVisible widgets with key [<'credential_input_form'>] descending from widgets with type "LoginView".
        ✅  13. enterText widgets with type "TextField" descending from widgets with key [<'login_username_input'>].
✅ Should see the same email when selecting filter and changing search input text (integration_test/tests/search/per

Test summary:
📝 Total: 1
✅ Successful: 1
❌ Failed: 0
⏩ Skipped: 0
📊 Report: file:///Users/datvu/WorkingSpace/tmail-flutter/build/app/reports/androidTests/connected/index.html
⏱️  Duration: 41s

Screen_recording_20250925_090703.webm

[github-actions]

This PR has been deployed to https://linagora.github.io/tmail-flutter/4057.

[tddang-linagora]

If there are many emails with attachments, will this test still find this email's subject?

[dab246]

If not found, it will be an error.