Incorrect preload & cause for the root's beforeLoad' arguments when defaultPreload: "intent" #4572

Which project does this relate to?

Router

Describe the bug

With defaultPreload: "intent" set on the router, the preload & cause arguments of the root's beforeLoad are incorrect when hovering Link.

There seems to be multiple things that are wrong here:

1. After initial load, hovering a Link causes beforeLoad to be triggered on the root with cause: enter - even on a link going to the currently active route. This behavior is inconsistent with the behavior after clicking a link, in which cause: stay is what's reported.
2. preload: false is always reported for the root's beforeLoad when hovering a Link. I would expect preload: true to be reported on a hover.
3. Going to a nested route, and then hovering on the link to the same route reports incorrect cause: "enter", preload: false on both the parent and on the nested route.

Your Example Website or App

https://stackblitz.com/edit/github-9xzbqucd?file=src%2Fmain.tsx

Steps to Reproduce the Bug or Issue

Reproduction for 1. above:

1. Hover the Home and About links.
2. Observe that the console reports the following:

root beforeLoad enter false        (expected "stay" - or maybe "preload"?)
index beforeLoad enter false      (expected "stay" - or maybe "preload"?)
root beforeLoad enter false        (expected "stay" - or maybe "preload"?)
about beforeLoad preload true

3. Click the Home link.
4. Hover the Home link.
5. Observe that the console reports the following:

root beforeLoad stay false
index beforeLoad stay false

Reproduction for 2. above:

1. Hover the About link.
2. Observe that the console reports root beforeLoad enter false (expected preload: true)

Reproduction for 3. above:

1. Click the Nested link.
2. Hover the Nested link.
3. Observe that the console reports the following:

root beforeLoad stay false        (expected `cause: "stay", preload: true` - or maybe `cause: "preload", preload: true`?)
bag beforeLoad enter false       (expected `cause: "stay", preload: true` - or maybe `cause: "preload", preload: true`?)
nested beforeLoad enter false  (expected `cause: "stay", preload: true` - or maybe `cause: "preload", preload: true`?)

Expected behavior

See notes in parentheses in the repro steps.

Screenshots or Videos

No response

Platform

• Router / Start Version: 1.124.0
• OS: Linux
• Browser: Chrome
• Browser Version: 138.0
• Bundler: vite
• Bundler Version: 6.3.5

Additional context

Related issues:

• preload="intent" triggers navigations in route's beforeLoad handler #1382