Problem

When you're in a chat, the browser sidebar sometimes pops open because a background pipe is driving the owned browser. It should only show for the chat that actually drove it.

Root cause

The owned browser is a single webview shared by every chat and every background pipe. Each navigation is tagged with an owner, and the sidebar ignores navigations owned by a chat other than the one on screen. That gate already worked for POST /navigate — it reads the x-screenpipe-session header the agent's curl shim injects (pipe:<name> for a pipe).

The gap: POST /eval with a url (navigate-and-scrape in one call — a common pipe path) never read that header. TauriOwnedHandle::eval hardcoded prepare_navigation(.., None), so the owned-browser:navigate event was emitted with owner = None, which the frontend treats as "the sidebar's own restore/reload" and always reveals.

Fix

Thread the owner through the eval path, mirroring the existing navigate_with_owner chain:

• browser_run_eval reads x-screenpipe-session and calls eval_with_owner
• new Browser::eval_with_owner / OwnedWebviewHandle::eval_with_owner (both default to plain eval), overridden by OwnedBrowser and TauriOwnedHandle
• TauriOwnedHandle::eval_inner threads owner into prepare_navigation

Defensive frontend tightening in isForeignNavigation: a tagged navigation with no bound chat (conversationId null — e.g. a brand-new unsaved chat) is now treated as foreign too, so a pipe can't pop into an empty chat.

Tests

• connect unit: eval_with_owner forwards the owner to the handle; plain eval stays un-owned
• engine integration: real axum router, POST /eval with x-screenpipe-session reaches the browser with the owner — this is the repro discriminator (pre-fix it routed through eval and the owner was None)
• frontend vitest: isForeignNavigation gates a tagged nav when no chat is bound
• e2e (zz-owned-browser-background-nav): drives the real /eval-with-url path with a foreign owner header and asserts the browser stays hidden

Local run: connect + engine cargo tests, frontend vitest, and bindings:check all green. The wdio GUI e2e runs in CI (not launched locally — there is a live screenpipe on :3030 it would conflict with).