Nonvanishing conjecture for the relevant p-adic L-functions

Let $E/K$ be the elliptic-curve setting of the paper, with $\pi$ and $\bar\pi$ denoting the relevant conjugate primes and $\chi$ and $\bar\chi$ the corresponding Galois characters. Consider the $p$-adic $L$-functions attached to the $\pi$-power- and $\bar\pi$-power-torsion of $E/K$.

Nonvanishing conjecture for the relevant p-adic L-functions. The $p$-adic $L$-function attached to the $\pi$-power-torsion does not vanish at negative powers of $\chi$, and the one attached to the $\bar\pi$-power-torsion does not vanish at negative powers of $\bar\chi$.

This conjectural nonvanishing is assumed, together with the Tate–Shafarevich conjecture, in the application of the Chabauty–Kim method; the source describes it as folklore and gives no resolution.