Palmieri's nilpotence conjecture for the power operation on Ext

Let $p$ be a prime, let $P_*$ be the polynomial part of the dual Steenrod algebra, and let $P$ be its dual. Consider $\operatorname{Ext}_P^{*,*}(\mathbb{F}_p,\mathbb{F}_p)$ and the algebra endomorphism

$$\mathcal{P}^0\colon \operatorname{Ext}_P^{s,t}(\mathbb{F}_p,\mathbb{F}_p)\longrightarrow \operatorname{Ext}_P^{s,pt}(\mathbb{F}_p,\mathbb{F}_p)$$

induced by the $p$th-power map on $P_*$. Palmieri's nilpotence conjecture. For fixed $s>0$, every $z\in(\mathcal{P}^0)^{-1}\operatorname{Ext}_P^{s,t}(\mathbb{F}_p,\mathbb{F}_p)$ is nilpotent. Equivalently, for every $z\in\operatorname{Ext}_P^{s,t}(\mathbb{F}_p,\mathbb{F}_p)$, there exists $n\geq0$ such that $(\mathcal{P}^0)^n z$ is nilpotent. The conjecture is proved when $p=2$, while the corresponding assertion for general primes remains open in the source.