One-dimensional representation conjecture for refined W-superalgebras

Let $\mathfrak{g}$ be a basic Lie superalgebra over $\mathbb{C}$, let $e\in\mathfrak{g}_{\bar 0}$ be nilpotent, and let $\chi$ be the corresponding linear function on $\mathfrak{g}_{\bar 0}$. Denote by $W_\chi'$ the refined $W$-superalgebra associated with $\chi$. Refined W-superalgebra representation conjecture. The refined $W$-superalgebra $W_\chi'$ affords a one-dimensional representation. This is a reformulation of the preceding conjecture using the structure theory of refined $W$-superalgebras. The general case is stated to remain open, with special cases established in the cited work.