Infinitely many failures of the cyclotomic unit condition

Determine whether, for a rank-two CM elliptic curve $E/\mathbb{Q}$ with $L(E,1)=0$ and root number $+1$, the set of split primes of good reduction outside $S_E$ at which the normalized second Taylor coefficient $\tilde c_2(p)$ is not a $p$-adic unit is infinite.

Background

For the CM rank-two curves studied in the paper, the unit condition $\tilde c_2(p)\in\mathbb{Z}_p^\times$ is equivalent, outside the excluded set, to the cyclotomic p-adic regulator being a unit together with triviality of the p-primary Shafarevich–Tate group.

Computations for five curves found only three failures among the relevant split primes below 30,000. The paper compares this with a random-residue heuristic predicting a very sparse but potentially infinite set of failures.