Communication problems for which a deterministic direct-sum theorem is not known to hold
I think I can suggest the problem, which is not widely known, but for which I'm interested in. Suppose we have $n$ permutations $\pi_i$ in $S_3$. Alice, Bob and Carol receive first, second and third elements of all permutations, respectively. The goal is to compute $\prod_i sgn(\pi_i)$, where $sgn(\pi_i)$ is a sign of permutation $\pi_i$ (can take -1 or +1). I'm interested in communication complexity in NiH model.

Some time ago I've tried to apply direct-sum theorem, but here we have some obstacles with dependence of input of different participants. I still don't know whether $\Omega(n)$ lower bound holds for this problem.