Mazur–Rubin–Sano exceptional zero conjecture for Rubin–Stark elements

Let $k_{\Gamma}/k$ be a $\mathbb{Z}_p$-extension with Galois group $\Gamma$, let $\Gamma_n:=\operatorname{Gal}(k_n/k)$, and put $L_\Gamma:=k_\Gamma L$. Let $S:=S_\infty(k)\cup S_p(k)\cup S_{\rm ram}(L/k)$, let $V:=\{v\in S\mid\chi(G_{k_v})=1\}$, and set $r:=\#V-g$. Writing $\mathcal A_\Gamma:=\ker(\mathbb{Z}_p^{\rm ur}[[\Gamma]]\to\mathbb{Z}_p^{\rm ur})$ for the augmentation ideal and $\mathbb T_\Gamma:=T\otimes_{\mathbb Z_p}\mathbb Z_p[[\Gamma]]$, define the inverse-limit Rubin–Stark element $\epsilon^{S_\infty(k),\chi}_{L_\Gamma/k,S}$ in ${\bigcap}^{g}_{\mathbb Z_p^{\rm ur}[[\Gamma]]}H^1(G_{k,S},\mathbb T_\Gamma)$. Mazur–Rubin–Sano exceptional zero conjecture.

$$\epsilon^{S_\infty(k),\chi}_{L_\Gamma/k,S}\in\mathcal A_\Gamma^r\cdot {\bigcap}^{g}_{\mathbb Z_p^{\rm ur}[[\Gamma]]}H^1(G_{k,S},\mathbb T_\Gamma).$$

The conjecture predicts that the inverse-limit Rubin–Stark element vanishes to order at least $r$ at the augmentation character, reflecting the exceptional zeros arising from primes in $V\setminus S_\infty(k)$.