Gras's Borel–Cantelli conjecture for p-rationality of real Galois fields

Let $K/\mathbb{Q}$ be a real Galois extension with Galois group $G$, and let $\eta\in E_K$ be a Minkowski unit, meaning a unit generating a sub-$G$-module of finite index in $E_K$. Let ${\mathcal R}_K$ be the normalized $p$-adic regulator, let

$$c_0(\eta)=\max_{\sigma\in G}|\eta^\sigma|,$$

and define $\log_2=\log\circ\log$. Gras's conjectural regulator estimate. The probability that ${\mathcal R}_K\equiv0\pmod p$ is at most

$$\frac{1}{p^{\log_2(p)/\log(c_0(\eta))-O(1)}}\quad\text{for }p\to\infty.$$

Under the principle of Borel–Cantelli, only finitely many primes $p$ satisfy that $K$ is non-$p$-rational. This is presented as a heuristic/conjectural formulation and the source gives no resolution.