Duplicated "Today" headers in timeline in a room and duplicate message sends

### Description

Multiple 'today's headers in a row.

The message that was sent just after the duplicated headers was sent again as I opened my laptop as well.

### Steps to reproduce

Unsure. Just reopened laptop from sleep after ~12h and got this. The duplicate today headers were only visible on my machine.

The duplicate message send is not local echo - other users on other homservers have seen it.

Logs being sent: yes


### Version information

- **Platform**: web

For the web app:

- **Browser**: Firefox 76.0.1
- **OS**:Ubuntu
- **URL**: riot.im/develop / riot.im/app / somewhere else? If a private server, what version of riot-web?

For the desktop app:

- **OS**: Windows, macOS, Ubuntu, Arch Linux, etc?
- **Version**: 1.x.y

Start by trying to reproduce the report in the web app with Firefox on Ubuntu after reopening the laptop from sleep, and review the submitted debug logs. The issue names no files or tests; done means a room timeline shows one Today header and a message is not sent again after wake-up.

Written by the indexing model from the issue text.