Auto hide isn't working on new searchbar

### Describe the bug

The new searchbar widget won't be hide when enable autohide nav-bar.

### Expected behavior

The searchbar is hide with the toolbar.

### Screenshots

_No response_

### OS

Windows 10

### OS - Others

_No response_

### Firefox Version

v149+

### Distribution

- [ ] [Original Lepton](https://github.com/black7375/Firefox-UI-Fix)
- [ ] [Lepton's photon style](https://github.com/black7375/Firefox-UI-Fix/tree/photon-style)
- [ ] [Lepton's proton style](https://github.com/black7375/Firefox-UI-Fix/tree/proton-style)

### Theme

- [ ] Light
- [ ] Dark
- [ ] Alpenglow (System: Light)
- [ ] Alpenglow (System: Dark)
- [ ] System Default
- [ ] Others: (e.g. [Photon Colors](https://addons.mozilla.org/en-US/firefox/addon/photon-colors/), [Google Chrome Dark](https://addons.mozilla.org/en-US/firefox/addon/google-chrome-dark/), [Fairytale Of Nature](https://addons.mozilla.org/en-US/firefox/addon/fairytale-of-nature/))

### Theme - More Info

_No response_

### `user.js` setup

user.js setup

```javascript
// Your settings
```

### Additional context

The new searchbar widget is a repurposed Url bar element.
So it need special treatment like url bar.

Workaround rule:
```css

/* fix new search widget auto hide */
@media -moz-pref("userChrome.autohide.tabbar") or -moz-pref("userChrome.autohide.navbar") {
@media -moz-pref("userChrome.autohide.navbar") or -moz-pref("userChrome.tabbar.one_liner") {
@media (prefers-reduced-motion: no-preference) {
#nav-bar:not([customizing]) #searchbar-new {
transition: opacity var(--uc-autohide-toolbar-speed) var(--animation-easing-function) var(--uc-autohide-toolbar-delay) !important;
}

#navigator-toolbox:is(:hover, :focus-within) #nav-bar:not([customizing]) #searchbar-new {
transition-delay: 0s !important;
}
}

@media -moz-pref("userChrome.autohide.navbar") {
@media not -moz-pref("userChrome.tabbar.one_liner") {
#navigator-toolbox:not(:hover, :focus-within) #nav-bar:not([customizing]) #searchbar-new {
pointer-events: none;
}

#nav-bar:not([customizing]) #searchbar-new {
opacity: 0;
will-change: margin-bottom, opacity;
}

#navigator-toolbox:is(:hover, :focus-within) #nav-bar:not([customizing]) #searchbar-new {
opacity: 1;
}
}
}
}
}

@media not -moz-pref("userChrome.urlView.as_commandbar") {
@media -moz-pref("userChrome.padding.urlbar") {
:root:not([uidensity="compact"]) #searchbar-new {
height: calc(var(--urlbar-height) - 2px) !important;
min-height: calc(var(--urlbar-min-height) - 2px) !important;
}
}
}
```

Start by finding the SCSS rules that implement navbar autohide and compare their treatment of the URL bar with the new #searchbar-new widget. Reproduce the issue on Firefox 149+ with userChrome.autohide.navbar enabled, then verify that the searchbar hides and reappears with the toolbar without breaking the provided workaround behavior.

Written by the indexing model from the issue text.