Cannot invite due to broken key backup

### Steps to reproduce

1. I created a private room on July 8 2026, e2ee enabled
2. I tried since multiple times to invite users to this room from different clients including SchildiNext and Element Desktop

### Outcome

#### What did you expect?

Invites work. They do using Nheko 0.12.1-d9189e58.

#### What happened instead?

Invites fail, claiming that "User does not exist". This is definitely wrong because they are local to my homeserver, which I also administrate, and I am actively sharing multiple rooms and DMs with them.

In devtools, I can see Element Desktop make 2 requests to `https:///_matrix/client/v3/room_keys/keys/?version=1`

### Operating system

Windows 10

### Application version

Element version: 1.12.23 Crypto version: Rust SDK 0.18.0 (6fe03c1), Vodozemac 0.10.0

### How did you install the app?

_No response_

### Homeserver

synapse v1.156.0

### Will you send logs?

Yes

Reproduce the invitation failure with Element Desktop 1.12.23 against Synapse 1.156.0, then inspect the two room_keys requests shown in devtools and collect the available logs. Compare the failing clients with Nheko 0.12.1-d9189e58; done means local users can be invited without the false “User does not exist” error.

Written by the indexing model from the issue text.