Query parameters in hash routing become permanent if used "as directed"

Describe the bug

In https://github.com/sveltejs/kit/pull/13191 there is a spirited discussion between Rich and the author as to how to handle query parameters in hash based routing.

Again, that's your choice. I will not; I will get my query parameters from page.url.searchParams, because I don't like to do work that's already been done for me.

Originally posted by @Rich-Harris in https://github.com/sveltejs/kit/issues/13191#issuecomment-2558122313

This implies that the intended "svelte" way to handle query parameters in hash based routing is with page.url.searchParams.

However -- when navigating between pages; the expected behaviour, ie, search parameters are deleted, does NOT occur. They become "permanent" search params.

Reproduction

https://stackblitz.com/edit/sveltejs-kit-template-default-h6zoggpw?file=src%2Froutes%2F%2Blayout.svelte

1. when opened, the footer says "You have no search params"
2. click about page - "Your current query parameters are: aboutpage="
3. click home - "Your current query parameters are: aboutpage="

Logs

System Info

System:
    OS: Linux 5.0 undefined
    CPU: (8) x64 Intel(R) Core(TM) i9-9880H CPU @ 2.30GHz
    Memory: 0 Bytes / 0 Bytes
    Shell: 1.0 - /bin/jsh
  Binaries:
    Node: 20.19.1 - /usr/local/bin/node
    Yarn: 1.22.19 - /usr/local/bin/yarn
    npm: 10.8.2 - /usr/local/bin/npm
    pnpm: 8.15.6 - /usr/local/bin/pnpm
  npmPackages:
    @sveltejs/adapter-auto: ^6.0.0 => 6.1.1 
    @sveltejs/kit: ^2.22.0 => 2.43.7 
    @sveltejs/vite-plugin-svelte: ^6.0.0 => 6.2.1 
    svelte: ^5.25.0 => 5.39.8 
    vite: ^7.0.4 => 7.1.8

Severity

serious, but I can work around it

Additional Information

No response

Reproduce the behavior in the linked StackBlitz example, starting with src/routes/+layout.svelte and the documented navigation steps. Trace how query parameters are read during hash-based navigation; done means moving from About to Home no longer retains aboutpage= when the destination has no search parameters.