[recheck after implementation] Using AdGuard Private Browser its uses Non-filtering AdGuard DNS configuration Protocol: DNS

### Please answer the following questions for yourself before submitting an issue

- [X] Filters were updated before reproducing an issue
- [X] I checked the [knowledge base](https://adguard.com/kb/) and found no answer
- [X] I checked to make sure that this issue has not already been filed

### AdGuard version

Version 4.7 Nightly 52

### Environment

- OS version: Android 14 One UI 6.1
- Device: Samsung Galaxy S24 Ultra

### HTTPS filtering

- [X] yes, I do

### Root access

- [ ] yes, I have it

### Integration with AdGuard VPN

- [X] yes, I do

### Routing mode

Local VPN

### Ad Blocking

AdGuard Base filter, AdGuard Mobile Ads filter

### Privacy

AdGuard Tracking Protection filter, AdGuard URL Tracking filter, EasyPrivacy

### Social

AdGuard Social Media filter, Fanboy's Social Blocking List

### Annoyances

AdGuard Cookie Notices filter, AdGuard Popups filter, AdGuard Mobile App Banners filter, AdGuard Other Annoyances filter, Adblock Warning Removal List

### Security

Online Malicious URL Blocklist

### Language-specific

AdGuard Chinese filter, AdGuard German filter, AdGuard Japanese filter, AdGuard Russian filter, AdGuard Ukrainian filter

### Other

_No response_

### Which DNS server do you use?

Custom DNS

### DNS protocol

DNS-over-QUIC

### Custom DNS

Adgaurd Personal DNS

### What Stealth Mode options do you have enabled?

Block trackers, Remove tracking parameters from URLs, Hide your search queries, Ask websits not to track you, Self-destruction of third-party cookies, Block ETag and If-None-Match headers, Block Push API, Block Location API, Hide your Referrer from third-parties, Hide your IP address, Remove X-client-Data header from HTTP request, Protect against DPI

### Issue Details

Steps to reproduce:
1. Login to AdGuard Personal DNS Account
2. Check test Page to find you want DNS Protocol and configuration is being used
3.

### Expected Behavior

Should use the configuration in my AdGuard Adblocker app which is set to Custom DNS from AdGuard Personal DNS QUIC

### Actual Behavior

Uses

Public server: running

You are using the Non-filtering AdGuard DNS configuration

Protocol: DNS

Server: chi-1

### Screenshots


### Additional Information

_No response_