Hofman-Zhang regulator divisibility conjecture for cyclic cubic fields

Let $p>3$ be a prime, let $K$ range over cyclic cubic fields, and let $R_{K,p}$ denote the normalized $p$-adic regulator. Hofman-Zhang's conjecture.

$$\mathrm{Prob}(p\text{ divides }R_{K,p})= \begin{cases} \dfrac{1}{p^2},&p\equiv 2\pmod 3,\\[6pt] \dfrac{2}{p}-\dfrac{1}{p^2},&p\equiv 1\pmod 3. \end{cases}$$

The source says that this prediction is supported by heuristic arguments and numerical experiments, while providing no proof or resolution.