Live Activity Push-to-Start does not work when the app is closed

## Live Activity Push-to-Start does not work when the Home Assistant app is closed

### Environment

* iPhone: iPhone 17 Pro
* iOS: 26.x
* Home Assistant iOS: 2026.9.2
* Home Assistant Core: 2026.9.2
* Installation: Home Assistant behind Nginx Proxy Manager
* External URL: HTTPS, accessible from the Internet

### Description

Live Activities work when the Home Assistant iOS application is open, but they never start when the application is completely closed.

This happens both on Wi-Fi and on cellular (4G/5G).

### Reproduction

With the Home Assistant app completely closed, I send this notification:

```yaml
action: notify.mobile_app_17_pro_quentin
data:
title: "Test Live Activity"
message: "Test"
data:
tag: test_live_5g
live_update: true
progress: 50
progress_max: 100
notification_icon: mdi:car-electric
```

Result:

* Home Assistant app open: Live Activity can be created.
* Home Assistant app closed: no Live Activity is created.
* Wi-Fi: no Live Activity when the app is closed.
* 4G/5G: no Live Activity when the app is closed.

### Important comparison

Normal remote notifications work correctly.

For example:

```yaml
action: notify.mobile_app_17_pro_quentin
data:
title: "Test 5G"
message: "Notification classique"
```

This notification is received immediately over 5G even when the Home Assistant app is closed.

Therefore:

* Remote push notifications: WORKING
* External HTTPS access: WORKING
* Nginx Proxy Manager: WORKING
* Live Activities permissions: ENABLED
* Live Activity synchronization: shows "Synchronized"
* Live Activity while app is active: WORKING
* Push-to-Start Live Activity while app is closed: NOT WORKING

### Local vs remote behavior

The notification history shows that Live Activities created locally appear as:

`Live Activity (Local)`

However, when the app is closed, no:

`Live Activity (Remote)`

entry appears.

Normal remote notifications are received correctly.

### Push-to-Start

The diagnostic logs show that a `start_live_activity_token` is generated and registered with Home Assistant.

Therefore, the Push-to-Start token appears to exist and is being registered, but it does not result in a Live Activity being started when the app is not running.

### Additional testing

I also tested:

* App Store version
* TestFlight version
* Wi-Fi
* 4G/5G
* Local Push enabled
* Live Activities enabled
* Frequent Live Activities enabled
* "Synchronize with Home Assistant" manually triggered
* New Live Activity tags
* Minimal Live Activity payload

The behavior remains the same.

### Related issues

This appears related to:

* #4907 — Live activities showing up as normal notifications
* #5459 — Live activity sometimes won't update or create

However, my specific reproducible behavior is that Push-to-Start does not create the Live Activity at all when the Home Assistant app is closed, regardless of whether the device is on Wi-Fi or cellular.

### Expected behavior

The Live Activity should be started by the Push-to-Start notification even when the Home Assistant app is completely closed, and should then appear on the Lock Screen / Dynamic Island.

### Actual behavior

Nothing happens until the Home Assistant application is opened.

Once the application is opened, the Live Activity can be created normally.

### Logs

I can provide diagnostics/logs if required.

Start with the Push-to-Start token registration and remote Live Activity delivery path, comparing behavior when the app is open versus completely closed. Reproduce with the provided YAML notification on Wi-Fi and cellular, then use the available diagnostics/logs to determine why no remote Live Activity appears; done means the activity starts and appears while the app is closed.

Written by the indexing model from the issue text.