The lower-central-series conjecture for generalized class of p-groups

Let $G$ be a finite $p$-group. Denote by $\gamma_i(G)$ the $i$th term of the lower central series of $G$, by $Cl(G)$ the generalized class parameter used in the paper, and by $K(G)$ the subgroup defined there. The lower-central-series conjecture.

$$\gamma_{Cl(G)+1}(G)\subseteq K(G).$$

This would strengthen the relationship between the generalized class parameter and the subgroup $K(G)$; the supplied text does not state whether the conjecture has been resolved.