Strengthened class-breadth conjecture for G-groups

Let $p>2$ be prime, let $G$ be a $p$-group, and let $P$ be a $G$-group. Let $\gamma_2^G(P)$ denote the second term of the $G$-lower central series, and let $\mathcal{M}^G(P)$ be the minimum integer $k$ such that the set of elements $x\in P$ with $G$-breadth $b^G(x)>k$ can be covered by two proper subgroups of $P$ containing $\gamma_2^G(P)$. A $G$-central series of $P$ is a series of $G$-invariant subgroups with successive terms as in the assertion.

Strengthened class-breadth conjecture. There exists a $G$-central series $C_i$, $0\leq i\leq N$, of $P$ such that

$$C_0=P,\qquad C_1=\gamma_2^G(P),\qquad C_N=\{e\},$$

and

$$\log_p[C_i:C_{i+1}]\leq\mathcal{M}^G(P)-i+1\qquad\text{for all }1\leq i\leq N-1.$$

This statement is presented as a generalization of the class-breadth conjecture for $p>2$-groups and of a previously known theorem. Its resolution is not established in the supplied text.