Cohomology stabilization conjecture for coclass families

Let $p$ be a prime, let $k$ be a field of characteristic $p$, and let $(G_i \mid i \geq 0)$ be a coclass family of finite $p$-groups. Cohomology stabilization conjecture. There exists $l \in \mathbb{N}$ such that

$$H^*(G_i,k) \cong H^*(G_j,k)$$

for all $i,j \geq l$. This conjecture extends the known result for finite $2$-groups to odd primes and strengthens it for the prime $2$; stabilization of the associated Quillen-category cohomology rings provides supporting evidence.