Depth-two conjecture for the finite p-groups $G_r=C_p\ltimes T_0/T_r$

Let $p$ be an odd prime and let $r\geq p-1$ be an integer. Let

$$G_r=C_p\ltimes T_0/T_r$$

be the finite $p$-group defined in the source. Depth-two conjecture. The mod-$p$ cohomology ring $\operatorname{H}^{*}(G_r;\mathbb{F}_p)$ has depth $2$. This is known for $p=3$ and $r=2$ or $r=3$, while the conjecture remains open in the generality stated. The source also notes that, for fixed $p$, the groups have isomorphic mod-$p$ cohomology groups as $\mathbb{F}_p$-modules, though not necessarily as rings.