Twisted-trace conjecture for Stark elements in abelian towers

Let $L/K/k$ be a tower of abelian extensions with groups $G=\operatorname{Gal}(L/k)$, $H=\operatorname{Gal}(L/K)$, and $\Gamma=\operatorname{Gal}(K/k)$. Let $S$, $T$, $S'$, and $S”$ be as in the source, with $r=|S'|$ and $s=|S”|-|S'|$. Assume that the Stark conjectures $\mathrm{St}(L/k,S,T,S')$ and $\mathrm{St}(K/k,S,T,S”)$ hold. Let $\operatorname{Tw}_{L/K}(m)=\sum_{h\in H}m^h\otimes h^{-1}$ be the twisted trace and let $\mathcal{A}_H$ be the augmentation ideal of $\mathbb{Z}[H]$.

Twisted-trace conjecture. The following two assertions hold:

1. $\operatorname{Tw}_{L/K}(\epsilon_L)\in\bigl(\bigwedge^rU_L\bigr)\otimes_G\bigwedge^rW^*_{L,S'}\otimes_{\mathbb{Z}}\mathcal{A}_H^s$;
2. modulo $\mathcal{A}_H^{s+1}$,

$$\operatorname{Tw}_{L/K}(\epsilon_L)\equiv(\mathbf{j}_{L/K}\otimes1)\bigl(\mathcal{R}^{\mathrm{Art}}_{L/K}(\epsilon_K)\bigr)$$

in $\bigl(\bigwedge^rU_L\bigr)\otimes_G\bigwedge^rW^*_{L,S'}\otimes_{\mathbb{Z}}\mathcal{A}_H^s/\mathcal{A}_H^{s+1}$. The conjecture relates leading terms in the equivariant class-number formula across an abelian tower and predicts both the order of vanishing and its leading coefficient. No resolution is supplied in the source or parser evidence.