It seems doable now, the first revision was aiming to support UHS-II also, but this feature was a little bit too much for the given space constraints so it was dropped. After that and some smart placement choices all parts fit in. PCB technology was estimated as 1+4+1 HDi, this stackup:

Simplified stack uses 4 layers standard process.

Unrouted signal, SD Card socket sense line. The rest is routed, there is some cleanup on the power layers to-do also.