Generalization of the eigencone and invariant restriction conjecture for symmetric subgroups

Let $G$ be a connected simply-connected, semisimple complex algebraic group, let $\sigma$ be a diagram automorphism of $G$, and let $K$ be its fixed subgroup. Fix a standard parabolic subgroup $P\subset G$ and Bruhat cells $\Lambda^P_{w_1},\ldots,\Lambda^P_{w_s}$ in $G/P$. For irreducible representations $V_{\lambda^1},\ldots,V_{\lambda^s}$ of $G$ with highest weights $\lambda^1,\ldots,\lambda^s$, define $\mathfrak h_K:=\mathfrak h^{\sigma}$ and let $\lambda^i_K$ be the restriction of $\lambda^i$ to $\mathfrak h_K$; these restrictions are dominant for $K$ with respect to $B^K:=B^{\sigma}$. The generalization conjecture. (a) There exist elements $k_1,\ldots,k_s\in K$ such that

$$\bigcap_{i=1}^s k_i\Lambda^P_{w_i}$$

is proper. (b) If

$$\left(V_{\lambda^1}\otimes\cdots\otimes V_{\lambda^s}\right)^G\neq 0,$$

then the tensor product of irreducible $K$-modules with highest weights $\lambda^1_K,\ldots,\lambda^s_K$ has a nonzero $K$-invariant. This conjecture is proposed as a common generalization of the preceding results for symplectic and odd orthogonal groups; the supplied passage does not state whether it has since been proved or disproved.