Common-solvability conjecture for systems of linear matrix inequalities

Let $A_1,\ldots,A_k$ and $B_1,\ldots,B_k$ be matrices of compatible sizes, and let $X$ range over Hermitian matrices. Consider the $k$ linear matrix inequalities

$$B_1XB_1^{*}\succcurlyeq A_1,\quad \ldots,\quad B_kXB_k^{*}\succcurlyeq A_k.$$

Common-solvability conjecture. The $k$ LMIs

$$B_1XB_1^{*}\succcurlyeq A_1,\quad \ldots,\quad B_kXB_k^{*}\succcurlyeq A_k$$

have a common Hermitian solution if and only if each of the $k$ LMIs has a Hermitian solution individually.

The preceding two-LMI result establishes this equivalence for $k=2$. The conjecture asks whether separate solvability always implies simultaneous solvability for an arbitrary finite family of LMIs of this form.