Ricci curvature rigidity conjecture for isoparametric tubes

Let $(N^{n+1},\mathbf{g})$ be a compact Riemannian manifold with boundary $\partial N$, and let $(M_c,\bar{\mathbf{g}}|_{M_c})$ be an isoparametric hypersurface in the unit sphere for some $c\in[c_0,1)$. Assume

$$\operatorname{Ric}_{\mathbf{g}}\geqslant n\mathbf{g}$$

on $N$, that $(\partial N,\mathbf{g}|_{\partial N})$ is isometric to $(M_c,\bar{\mathbf{g}}|_{M_c})$, and that $\mathrm{II}_{\mathbf{g}}\geqslant\mathrm{II}_{\bar{\mathbf{g}}}$ pointwise on $\partial N\cong M_c$. Ricci curvature rigidity conjecture. Then $(N^{n+1},\mathbf{g})$ is isometric to the isoparametric tube $(\mathbb{S}^{n+1}_{F+c},\bar{\mathbf{g}})$ in $\mathbb{S}^{n+1}$. The hemisphere case $g=1$ is known, and the $g=2$ case was subsequently verified; the general case remains open.