restore the activity stamp across restarts: the quiet ordering no longer flattens on deploy

Root cause of the reviewer's mass-bump measurement (17 of 17 sessions with an
identical lastActivityAt): every restart restamps all sessions in the
constructor loop, and the boot auto-attach's repaint re-bumps the rest within
the same second. A 12-minute steady-state sample shows NO ambient mass bump,
so restarts are the whole story, and Codeman restarts on every deploy.

The stamp now has a display twin: recovery threads the previous run's
lastActivityAt from state.json into the wire-visible stamp (getter + toState),
and a 15s settle window keeps the attach repaint from overwriting it. Real
actions (input, task assignment, respawn) always write through. The private
stamp keeps its boot-anchored semantics untouched, because the idle
confirmation reads it as how long the pane has been quiet.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

src/web/server.ts

@@ -2657,6 +2657,11 @@ export class WebServer extends EventEmitter {
               // the launch conversation until the user types again, even though
               // the re-attached CLI is on a post-`/clear` one.
               lastSubmitAt: savedState?.lastSubmitAt,
+              // The pane's last output, previous run's value. Without it every
+              // restart restamped all sessions "now" (constructor + the attach
+              // repaint within the same second), flattening the home screens'
+              // most-recently-quiet ordering to tab order after each deploy.
+              lastActivityAt: savedState?.lastActivityAt,
               // Remote SSH metadata must round-trip on recovery: without it the
               // attach cwd falls back to the (nonexistent-locally) remote path and
               // respawn rebuilds a LOCAL command, breaking the pane and silently