The polylogarithmic-party interleaved group-product hardness conjecture

Let $G$ be a non-abelian simple group, let $c>0$ be a constant, and let $k=\log^c t$. The $k$-party $t$-tuple interleaved group product over $G$ is the problem of computing the interleaved product of the group elements in the tuple.

Polylogarithmic-party hardness conjecture. There is no protocol for the $k$-party $t$-tuple interleaved group product over $G$ with $k=\log^c t$ parties and communication $\log^c t$.

This more concrete formulation appears in a false-commented-out portion of the source, so its status in the paper's active argument is unclear. The provided text does not indicate whether it has been resolved.