Add Open in Element chat button to event detail page

As an Event Participant, I want a button to open the event's Matrix chat room in Element, so I can communicate with other attendees without leaving the browser.

### Acceptance Criteria
- [ ] A chat button is shown on the event detail page for participants only (the current user must have joined the event)
- [ ] The button opens https://matrix.to/#/{matrixRoomId} in a new tab (test the URL with Leo beforehand)
- [ ] The button is not shown if the event has no matrixRoomId
- [ ] The button is not shown if the current user is not a participant

Start by locating the event detail page and the participant and matrixRoomId data used there. Add or verify the chat button against all three visibility conditions, confirm it opens the specified Matrix URL in a new tab, and test the link with Leo before checking participant and missing-room cases.

Written by the indexing model from the issue text.