Kahn's Stiefel–Whitney conjecture for finite 2-groups

Throughout, let $G$ be a finite $2$-group, let $d(G)$ denote its minimal number of generators, and let $r_G$ be the real regular representation of $G$. Define

$$\nu(G)=\min\{n>0\mid w_{2^{n-1}}(r_G)\neq 0\},$$

where $w_i(r_G)$ is the $i$-th Stiefel–Whitney class of $r_G$. Kahn's conjecture.

$$\nu(G)\geq d(G).$$

This conjecture connects the minimal number of generators of a finite $2$-group with the first nonzero Stiefel–Whitney class of its real regular representation. The supplied text gives no evidence of a resolution.