The one-step nilpotence conjecture for Palmieri's power operation

Let $p$ be a prime and let $\mathcal{P}^0$ be the endomorphism of $\operatorname{Ext}_P^{*,*}(\mathbb{F}_p,\mathbb{F}_p)$ induced by the $p$th-power map on the polynomial part of the dual Steenrod algebra. One-step nilpotence conjecture. For every $z\in\operatorname{Ext}_P^{s,t}(\mathbb{F}_p,\mathbb{F}_p)$, the element $\mathcal{P}^0z$ is nilpotent. This strengthens Palmieri's preceding conjecture by asserting that one may take $n=1$; the source says it is suggested by hand and computer calculations when $p=2$, and does not state a resolution.