Tao's asymptotic tightness conjecture for colored product-free subsets

Let $G$ be a finite group, let $\mathbb{F}$ be a field, and let $X_1,\dots,X_k$ be subspaces of the group algebra $\mathbb{F}[G]$ satisfying

$$X_1\cdots X_k=0.$$

Define $M_k(G)$ to be the minimum, over all choices of the field and all such $k$-tuples of subspaces, of

$$\sum_{i=1}^k \operatorname{codim} X_i.$$

A $k$-colored product-free subset of $G$ is a set of $k$-tuples with no product relation of the relevant colored form. Tao's asymptotic tightness conjecture. For sufficiently large $|G|$, the maximal cardinality of a $k$-colored product-free subset of $G$ is

$$M_k(G)\cdot |G|^{o(1)}.$$

The preceding theorem gives the upper bound by the sum of codimensions, while the conjecture asserts that this bound is asymptotically sharp up to a subpolynomial factor. Tightness is known in logarithmic scale for tri-colored sum-free subsets of $\mathbb{F}_p^n$, from work of Kleinberg, Sawin, Speyer, Norin, and Pebody; the corresponding assertion for other groups is posed as an open problem.