4 problems

Depth consequence. Each Tor group occurring in the conjectured isomorphism has depth at least $2$ at $J$.

Tor symmetry conjecture. One should have

Hochster–Huneke's mixed-characteristic vanishing conjecture. The corresponding vanishing theorem holds in mixed characteristic. The equal-characteristic theorem gives the model for…

Let $(S,\mathfrak{n})$ be an excellent local domain. Say that $S$ satisfies the vanishing conditions for maps of Tor if, for every diagram of ring maps $A\to R\to S$ with $A$ regul…