The Pfaffian generator conjecture for truncated shifted twisted Yangians

Let $N$ be even, let $M=2m$ be the number of boxes in the associated pyramid, and let ${\mathscr{Y}}_{N,\ell}^{\pm}(\sigma)$ be a truncated shifted twisted Yangian. A Pfaffian generator is a central element $\mathpzc{Pf}\in {\mathscr{Y}}_{N,\ell}^{\pm}(\sigma)$ of canonical degree $m$ such that $z_2,\ldots,z_{2m-2},\mathpzc{Pf}$ are algebraically independent.

Pfaffian generator conjecture. A Pfaffian generator exists if and only if either ${\mathscr{Y}}_{N,\ell}^{+}(\sigma)$ has $N$ even and $\ell$ odd, or ${\mathscr{Y}}_{N,\ell}^{-}(\sigma)$ has $\ell$ even.

The only-if direction follows from the theorem on the truncated center. Brown's work supports existence in all unshifted cases, and the source proves existence for ${\mathscr{Y}}_{2,\ell}^{+}(\sigma)$ when $\ell$ is odd; the general existence direction remains open.