Unitarity conjecture for exceptional W-algebras of type d535_k(\mathfrak{sp}4,f{subreg})

Let $k$ be one of the four levels

$$k\in\left\{-\frac{5}{3},-\frac{4}{3},-\frac{7}{4},-\frac{5}{4}\right\}.$$

The vertex algebra under consideration is the exceptional W-algebra associated with the subregular nilpotent element $f_{subreg}$ of $\mathfrak{sp}_4$. Unitarity conjecture. At these levels, the vertex algebra

$$\mathcal{W}_k(\mathfrak{sp}_{4},f_{subreg})$$

is unitary. The preceding discussion establishes positivity at these levels, while unitarity would additionally require nonnegative conformal dimensions for the Virasoro module structure; the conjecture asserts this stronger property.