New dashboard checks have stale state when tests are queued

### Help us understand the severity of this issue

- [ ] breakage; unable to contribute or trigger builds will no workarounds
- [ ] inconvenient workarounds exist, but significant effort required
- [ ] needed for a Flutter team-wide priority (already agreed upon)
- [x] nice-to-have; none of the above

### What do you need help with

Two screenshots show the "current" as of the time of filing state for https://github.com/flutter/packages/pull/11943. It shows dashboard checks as green from the previous run while the actual state is queued.
It looked like based on the github ui that I could have merged if I had approvals.

Start with flutter/packages PR #11943 and compare the dashboard checks shown in the screenshots with the checks' actual queued state. Trace where the dashboard obtains and refreshes check results; done means a queued check no longer displays the previous run as green.

Written by the indexing model from the issue text.