Shimizu–Hirakawa conjecture on coefficients of fundamental units

Let $p\neq 3$ be a prime number, let $P=p$ or $2p$, and let

$$\epsilon=\alpha+\beta p^{1/3}+\gamma p^{2/3}\in\mathbb{R}_{>1}$$

be the fundamental unit of $\mathbb{Q}(P^{1/3})$, where $\alpha,\beta,\gamma\in(1/3)\mathbb{Z}$. Shimizu–Hirakawa conjecture. We have

$$\beta\not\equiv 0\pmod p.$$

This conjecture would imply that the parameter $\iota$ used in the construction of non-singular plane curves violating the local-global principle equals $1$ for every prime $p\neq 3$. Its status is not resolved in the supplied source.