Quantum communication upper-bound conjecture for Hidden Hypermatching

Let $r,t>2$, and let $r\text{-}{\mathsf{HH}}(\alpha,t,n)$ denote the $r$-ary Hidden Hypermatching problem with parameters $\alpha$, $t$, and $n$. A protocol uses quantum communication measured in qubits.

Quantum Hidden Hypermatching upper-bound conjecture. If $t,r>2$, there is a protocol for $r\text{-}{\mathsf{HH}}(\alpha,t,n)$ using

$$O\left(\log(rn)(n/\alpha)^{1-1/\lceil t/2\rceil}\right)$$

qubits.

The conjectured bound would improve the known quantum upper bound in the regime $t,r>2$ and match the corresponding classical communication complexity. The surrounding discussion presents it as motivated by the lower bounds, while the general case remains unresolved.