Finite-dimensionality criterion for irreducible twisted super Yangian modules

Let ${\mathscr{B}_{\bm s,\bm \varepsilon}}$ be the twisted super Yangian of type AIII, let $V(\bm\mu(u))$ be an irreducible ${\mathscr{B}_{\bm s,\bm \varepsilon}}$-module, and let $\varkappa$ denote the rank parameter. An $\ell_{\bm s}$-weight is a tuple $\bm\lambda(u)=(\lambda_i(u))_{1\leqslant i\leqslant \varkappa}$ of formal series of the prescribed type. Finite-dimensionality criterion. If $V(\bm\mu(u))$ is finite-dimensional, then there exist $\gamma\in\mathbb{C}$ and an $\ell_{\bm s}$-weight $\bm\lambda(u)$ such that the equations referred to as hold and the corresponding ${\mathscr{Y}_{\bm s}}$-module $L(\bm\lambda(u))$ is finite-dimensional. The preceding result proves the sufficiency of these conditions; this statement asserts their necessity and is presented as an expectation, so its status is open. This candidate depends on equation, whose explicit form is not included in the supplied excerpt.