The slice-rank conjecture for group multiplication tensors

Let $G$ be a finite group, let $M(G)$ denote its matching number, and define the minimum slice rank

$$\operatorname{SR}(G)=\min_k \operatorname{SR}(T_{k[G]}),$$

where the minimum is over algebraically closed fields $k$ and $T_{k[G]}$ is the multiplication tensor of the group algebra $k[G]$. The slice-rank conjecture. One has

$$\operatorname{SR}(G)\leq M(G)\cdot |G|^{o(1)}.$$

This would make the minimum slice rank of the group multiplication tensor essentially controlled by the largest matching in the group and would yield corresponding slice-rank bounds for matching problems in groups. The source describes this as a conjecture slightly weaker than one previously proposed by Petrov; no resolution is supplied here.