fix: ON AIR badge correct at launch + faster to appear

At launch nothing is in memory yet, so the badge showed offline even with a
recent QSO. liveLastQSOTime is now authoritative: it takes the most recent of
the in-memory stamp AND the DB (this operator's last QSO — covers a contact
from the shared logbook or one logged before launch), used by both the
published status and the badge. The badge polls the backend every 5 s (down
from 15) and on each qso:logged, so it shows on air within a few seconds and
flips online instantly on a new contact.

livestatus.go

@@ -87,12 +87,29 @@ func (a *App) seedLiveLastQSO() {
 	}
 }
 
-// LiveLastQSOAgeSec returns seconds since this operator's last logged QSO, or -1 if
-// none is known — the UI uses it to seed the "on air" badge at launch.
-func (a *App) LiveLastQSOAgeSec() int {
+// liveLastQSOTime is the authoritative "last contact" instant for this operator:
+// the most recent of the in-memory stamp (this session's local logs, updated
+// instantly) AND the DB (a contact that arrived via the SHARED logbook from another
+// station, or one logged before launch). Used by both the published status and the
+// UI badge so on-air/offline is right in every multi-op case.
+func (a *App) liveLastQSOTime() time.Time {
 	a.liveActMu.Lock()
 	last := a.liveLastQSOAt
 	a.liveActMu.Unlock()
+	if a.qso != nil {
+		if op, _ := a.liveStatusOperator(); op != "" {
+			if t, ok := a.qso.LastQSOTime(a.ctx, op); ok && t.After(last) {
+				last = t
+			}
+		}
+	}
+	return last
+}
+
+// LiveLastQSOAgeSec returns seconds since this operator's last logged QSO, or -1 if
+// none is known — the UI polls it for the "on air" badge.
+func (a *App) LiveLastQSOAgeSec() int {
+	last := a.liveLastQSOTime()
 	if last.IsZero() {
 		return -1
 	}
@@ -182,8 +199,8 @@ func (a *App) publishLiveStatus() {
 	if mode == "" {
 		mode = a.liveMode
 	}
-	lastQSO := a.liveLastQSOAt
 	a.liveActMu.Unlock()
+	lastQSO := a.liveLastQSOTime() // authoritative (in-memory OR shared DB)
 	// Online = a new contact was logged within the window. An operator who leaves
 	// the log open but stops working shows offline after `liveOnlineWindow`; the
 	// next QSO flips them back on. never-logged (zero time) → offline.