Density and rank-stability conjectures for generalized non-Wieferich primes

Let $E/\mathbb{Q}$ be an elliptic curve and let $P\in E^0(\mathbb{Q})$ be a non-torsion point. Define the sets of good primes

$$\Lambda_{E,P}=\{p:q_p(a_{N_p})\neq0\}=\{p:a_{N_p}^{p-1}\not\equiv1\pmod {p^2}\},$$

and $W_{E,P}$ as the set of primes for which $N_pP\not\equiv0\pmod {p^2}$, where $N_p$ is the number of points of the reduction of $E$ modulo $p$. The generalized non-Wieferich and rank-stability conjectures. The set $\Lambda_{E,P}$ has density $1$ and an infinite complement; the set $W_{E,P}$ has density $1$ and an infinite complement; and, for every elliptic curve $E/\mathbb{Q}$ of rank $1$, only finitely many supersingular primes $p$ cause the rank to change in the cyclotomic $\mathbb{Z}_p$-extension. These claims would explain the observed rarity of simultaneous exceptional signed Iwasawa behavior and rank growth in the supersingular cyclotomic tower. The supplied text gives no resolution status.