consult-recent-file list not getting updated in daemon mode

### Describe your issue

When I open a file, close the emacs client and then restart the daemon, the opened file is NOT in consult-recent-file.
When I open a file, close the emacs client and open the emacs client again without restarting the daemon, the file IS in consult-recent-file.

edit:
I'm reporting this here instead of for upstream consult.el because I'm sure I read in some doom documentation, that the dashboard page is supposed to handle the saving of the recent files.
If this assessment is wrong, I will report the bug upstream.
I'm using the new dashboard btw. Doom info confirms this.

### Steps to reproduce

1. start emacs daemon
2. start emacs client
3. open file x (any file that is not in consult-recent-file yet)
4. stop emacs client
5. stop emacs daemon
6. start emacs daemon
7. start emacs client
8. run consult-recent-file
9. notice how file x is not in the list

### System information

https://pastebin.com/kw03eXQh

### Disclosures

- [ ] This issue was written with/by AI.

Reproduce the daemon/client sequence and inspect how the new dashboard handles saving recent files around consult-recent-file. Compare restarting the daemon with only restarting the client, then trace the relevant dashboard and recent-file entry points. Done means a file opened in a client remains available in consult-recent-file after the daemon is restarted.

Written by the indexing model from the issue text.