It doesn't make sense for speculatively executed code to throw architecturally visible exceptions. The appropriate behavior would be to not perform speculative loads across protection domains (i.e. the behavior of AMD implementations).

It would make sense if it was the only alternative as the kernel can handle it. The appropriate behavior is to remove all traces of the speculative execution including cache hits.

Is that even possible? The data that would need to be removed from the cache has already evicted other cache lines, and that re-fetching those might have observable effects, like the timing.