The field-extension invariance conjecture for degree sequences of regular sequences

Let $G$ be a finite $p$-group, let $k$ be a positive integer, and let $d$ and $a_1,\dots,a_d$ be integers. A regular sequence has degree sequence $a_1,\dots,a_d$ when its elements have respective degrees $a_1,\dots,a_d$. The field-extension invariance conjecture. If $H^*(G,\mathbb F_{p^k})$ admits a regular sequence of length $d$ with degree sequence $a_1,\dots,a_d$, then $H^*(G,\mathbb F_p)$ admits a regular sequence of equal length and with the same degree sequence. This is a proposed descent statement for regular sequences along the finite field extension $\mathbb F_p\subseteq\mathbb F_{p^k}$. Jon F. Carlson predicted that the conjecture would be false for many $p$-groups; the supplied text does not give a proof or explicit counterexample.