Gross's refined class number formula conjecture

Let $K'$ be the relevant extension, let $H'$ be its Galois group, let $I(H')$ be the augmentation ideal, and let $r_{K'}$ be the rank of the $S(K')$-unit group. Let $\theta_{H'}$ be the Stickelberger element, let $h_{K',S(K'),T(K')}$ be the modified class number, and let $\det_{H'}$ be Gross's refined regulator. Gross's conjecture. One has $\theta_{H'}\in I(H')^{r_{K'}}$ and

$$\theta_{H'}\equiv h_{K',S(K'),T(K')}\cdot \det_{H'}\pmod{I(H')^{r_{K'}+1}}.$$

This conjecture refines the class number formula by identifying the leading term of the Stickelberger element with Gross's refined regulator. The source discusses it as an ingredient for proving the paper's theorem but does not state its resolution.