The quasirandom groups conjecture for modular group algebras

Fix $\varepsilon>0$. Let $G$ be a finite group of order $n$, and say that $G$ is $n^\varepsilon$-quasirandom when every nontrivial irreducible complex representation of $G$ has dimension at least $n^\varepsilon$. For a field $k$, let $\operatorname{J}(k[G])$ denote the Jacobson radical of the group algebra $k[G]$. The quasirandom groups conjecture. For all fields $k$, one has the uniform bound

$$\dim k[G]/\operatorname{J}(k[G])\geq \Omega(n).$$

This conjecture asks whether quasirandomness over the complex numbers forces the semisimple quotient of the group algebra to have linear dimension uniformly in every characteristic. The statement is presented as an open question motivated by the preceding calculation for $\operatorname{PSL}(2,p)$, and no resolution is given.