Neveu–Schwarz unitarity conjecture for minimal W-algebra highest weight modules

Let $W^k_{\min}(\mathfrak g)$ be the minimal $W$-algebra, let $P^+_k$ denote the relevant set of dominant integral weights, and let $A(k,\nu)$ be the lower bound for the conformal weight. Write

$$\left\{L^W(\nu,\ell)\mid \nu\in P^+_k,\ \nu\text{ non-extremal},\ \ell\ge A(k,\nu)\right\}$$

for the already known unitary highest weight modules, where extremality means that $\nu(\theta_i^\vee)>M_i(k)+\chi_i$ for some $i$. Neveu–Schwarz unitarity conjecture. The set of unitary highest weight modules over $W^k_{\min}(\mathfrak g)$ is the union of

$$\left\{L^W(\nu,\ell)\mid \nu\in P^+_k,\ \nu\text{ non-extremal},\ \ell\ge A(k,\nu)\right\}$$

and

$$\left\{L^W(\nu,A(k,\nu))\mid \nu\in P^+_k,\ \nu\text{ extremal}\right\}.$$

The non-extremal part is a theorem, while the conjecture asserts unitarity and completeness for the extremal boundary modules.