[POP Keys]: Emoji Buttons not Mapping

### Pre-flight checklist

- [x] I quit **Logi Options+** before testing.
- [x] I searched existing issues for my device.

### Device model

Logi POP Keys

### Connection

Logi Bolt receiver

### What's the problem?

Device is detected but a feature doesn't work

### `openlogi list` output

```text
(inventory read from the running agent)
Logi Bolt Receiver (6C86A59430BD22FD, vid=046d pid=c548)
└─ slot 2 ● Logi POP Keys (keyboard, wpid=b365, battery=100% full (discharging))
model_ids=[b365,0000,0000] ext=03 serial=2242GT0YL5B8 unit_id=0c7728ec transports=btle
```

### `openlogi diag features` output

```text
device: Logi POP Keys (slot 2 on receiver 6C86A59430BD22FD)
idx id ver flags
0 0x0000 v2
1 0x0001 v2
2 0x0003 v4
3 0x0005 v0
4 0x1d4b v0
5 0x0020 v0
6 0x0007 v0
7 0x1004 v3
8 0x1b04 v5
9 0x1814 v1
10 0x1815 v2
11 0x40a3 v0
12 0x4220 v0
13 0x4521 v0
14 0x4531 v1
15 0x4540 v0
16 0x00c3 v0
17 0x1802 v0 hidden,engineering,manufacturing-deactivatable
18 0x1803 v0 hidden,engineering,manufacturing-deactivatable
19 0x1806 v8 hidden,engineering,manufacturing-deactivatable
20 0x1816 v0 hidden,engineering,manufacturing-deactivatable
21 0x1805 v0 hidden,engineering
22 0x1830 v0 hidden,engineering,manufacturing-deactivatable
23 0x1891 v7 hidden,engineering,compliance-deactivatable
24 0x18a1 v0 hidden,engineering,manufacturing-deactivatable
25 0x1e00 v0 hidden
26 0x1e02 v0 hidden,engineering
27 0x1602 v0
28 0x1eb0 v0 hidden,engineering,manufacturing-deactivatable
29 0x1861 v1 hidden,engineering,manufacturing-deactivatable
30 0x18b0 v0 hidden,engineering,manufacturing-deactivatable
(31 feature entries)

fw 0: Bootloader BL144.01_B0008 pid=b365 extra=2be8baf400
fw 1: MainApplication RBK69.01_B0008 pid=b365 [active] extra=2be8baf400
fw 2: Softdevice 00.00_B0119 pid=0000
```

### Details

I'm on Windows 11 with no admin rights. However the diag features output also fails when running an elevated shell as well.

I'm not able to map the weird emoji keys down the right side of the keyboard. In the UI, the F13-F19 controls don't actually connect to any buttons visually. Assigning actions to any F keys doesn't actually do anything. I can't tell if there's something not working with the communications, or if the device is just broken in the app.

_Task Manager shows no Logi Options running_

_FKey icons don't have a working mapping_

I can see all the tasks with openlogi diag controls, so at least something is working. Any ideas?

```
device: Logi POP Keys (slot 2 on receiver 6C86A59430BD22FD)
cid task flags capabilities
0x00d1 0x00ae 0x040a analytics-events
0x00d2 0x00af 0x040a analytics-events
0x00d3 0x00b0 0x040a analytics-events
0x006e 0x0043 0x043a divertable, analytics-events
0x010a 0x00df 0x043a divertable, analytics-events
0x011c 0x00f1 0x043a divertable, analytics-events
0x00e4 0x0006 0x043a divertable, analytics-events
0x00e5 0x0004 0x043a divertable, analytics-events
0x00e6 0x0005 0x043a divertable, analytics-events
0x00e7 0x0003 0x043a divertable, analytics-events
0x00e8 0x0002 0x043a divertable, analytics-events
0x00e9 0x0001 0x043a divertable, analytics-events
0x0103 0x00d8 0x0434 divertable, analytics-events
0x0104 0x00d9 0x0434 divertable, analytics-events
0x0105 0x00da 0x0434 divertable, analytics-events
0x0106 0x00db 0x0434 divertable, analytics-events
0x0107 0x00dc 0x0434 divertable, analytics-events
0x0108 0x00dd 0x0434 divertable, analytics-events
0x00de 0x0062 0x0402 analytics-events
0x0034 0x0062 0x0404 analytics-events
```

Reproduce the issue with the Logi POP Keys and run `openlogi diag features` and `openlogi diag controls` to compare the reported features and tasks. Trace how the F13-F19 emoji buttons are discovered and presented in the remapping UI. Done means the emoji buttons appear as selectable controls and assigned actions respond when pressed.

Written by the indexing model from the issue text.