Generalization of Reichstein–Youssin's fixed-point theorem

Let $k$ be the base field, let $d$ be a positive integer, and let $G$ be a finite group with subgroups $H_1,\dots,H_r$. Write $\operatorname{rdim}_k(H_i)$ for the representation dimension of $H_i$ over $k$.

Generalized fixed-point conjecture. If $\operatorname{rdim}_k(H_i)\leq d$ for all $i=1,\dots,r$, then there exists a $d$-dimensional $k$-variety $X$ with a faithful $G$-action and smooth $k$-points $x_1,\dots,x_r\in X$ such that $H_i$ fixes $x_i$ for each $i=1,\dots,r$.

This is proposed as a generalization of Reichstein–Youssin's theorem on constructing varieties with prescribed subgroup-fixed smooth points. The supplied text gives no resolution, so the conjecture remains open.