Firebase doesn't respect Next.js error pages

As described in the summary,
when serving Firebase+Next project build with `firebase serve` or `firebase emulators:start --only hosting`,
Firebase shows general 'Not found' or 'Cannot GET ...' errors with no styling instead of generated by Next.js 404 page.

*STR*
1. `firebase init` -> hosting + emulator for hosting
2. `firebase serve` or `firebase emulators:start --only hosting`
3. Visit localhost:5000/whatever

**AB**

**EB**
Error page should be the same as when you run `next dev`

**Env:**
node: v16.19.0
firebase-tools: 11.22.0
next: 13.1.6