Conjecture on the periodic discriminant equality for real cubic fields

Let $K$ be a real cubic field and let $r=5/2$. Write $\Delta_K$ for the discriminant of $K$ and $\Delta_{K,r}^{\mathcal{P}er}$ for the associated periodic discriminant quantity. Periodic discriminant conjecture. For every real cubic field $K$,

$$\Delta_K=\Delta_{K,r}^{\mathcal{P}er}.$$

This conjecture is presented as being supported by numerical experiments; the supplied text gives no resolution or further context.