Persist per-screen selection and search/filter across tab navigation; clear only on user action or disconnect #1417

Background / change of direction

Each screen currently keeps its selection and filter state local, so switching tabs unmounts the screen and discards it. The Tools screen even goes a step further: because the call result lives in App (so it survives pending → ok/error), ToolsScreen clears the result on unmount to avoid showing a result with no selected tool (see the onClearResult unmount effect / comment in ToolsScreen.tsx).

We're reversing that direction. Navigating away from a screen and back should retain what the user was doing. Session state should be cleared only:

• explicitly by the user (a clear/reset affordance), or
• on disconnect (the session ends).

Scope — what should persist across navigation

• Tools — selected tool, its form values, and its result
• Prompts — selected prompt, its argument values, and its result
• Resources — selected resource and its read result (resource subscriptions already live in managed state)
• Apps — selected app, its form values, and running/maximized state
• Tasks — selected task
• Logs — search text + level filters
• History — search text + method filter
• Network — search text + category filters

Desired behavior

• Selecting/typing on a screen, switching tabs, and returning shows the same selection / result / search / filters.
• A result is never shown without its originating selection (the bug the unmount-clear was guarding against) — because the selection now persists alongside the result.
• State is cleared on disconnect and via an explicit user action; nothing clears it on plain tab navigation.

Implementation sketch

• Lift the per-screen selected* / formValues / search / filter state out of the screen components into session-scoped state owned by App (alongside toolCallState / getPromptState / readResourceState), passed down as props.
• Clear it in App's resetSessionScopedUiState() (already invoked on the InspectorClient disconnect event — see App.tsx), and from the explicit user clear action(s).
• Remove ToolsScreen's clear-on-unmount effect — it's superseded once selection persists.
• Keep each screen a controlled "dumb" component (state + callbacks via props), consistent with the v2 architecture.

Relationship to #1414

#1414's companion criterion ("persist tool selection + results across navigation") is the Tools-only slice of this. This issue generalizes it to every screen plus search/filters. Suggest implementing the persistence here and trimming #1414 down to its toasts-for-progress scope (or closing the overlap there) so the two don't conflict.

Acceptance criteria

• Each screen above retains its selection / result / search / filters across tab navigation within a live session.
• No screen clears that state on plain navigation.
• Disconnect clears all of it; an explicit user action clears it too.
• ToolsScreen's unmount result-clear is removed.
• Tests cover persist-across-navigation and clear-on-disconnect for the lifted state.