postMessage cannot provide a reliable timing signal since it goes on the task queue on the receiving end (in the main thread) along with other pending events, and even if there were no other events, there is latency noise in postMessage due to the fact that the web worker is not the only thread running on the CPU. Some suggest that the attack would only take more time as the attacker has to collect a bigger sample, and factor out the noise, but I haven't seen a public exploit based on that.

The other angle of attack that used to be viable was documented in this HN comment: https://news.ycombinator.com/item?id=14057091

But AFAIK all browsers have disabled SAB, e.g. see: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Refe...

EDIT:

Chrome has re-enabled SAB, with mitigations.