Strong rationality equivalence for unitary minimal W-algebras

Let $W_k(\mathfrak{g},f)$ be a unitary minimal $W$-algebra, with corresponding conformal net denoted by $\mathcal{A}_{W_k(\mathfrak{g},f)}$. The algebra is strongly rational when it has the strong rationality properties used in the paper, and the conformal net is completely rational in the sense of conformal-net theory. Strong rationality equivalence conjecture.

$$W_k(\mathfrak{g},f)\text{ is strongly rational}\quad\Longleftrightarrow\quad\mathcal{A}_{W_k(\mathfrak{g},f)}\text{ is completely rational}.$$

The theorem preceding this conjecture proves the equivalence for $\mathfrak{g}\in\{\mathfrak{sl}_2,\mathfrak{spo}(2|1),\mathfrak{spo}(2|2),\mathfrak{spo}(2|3),\mathfrak{psl}(2|2)\}$ or at a collapsing level, while the conjecture asks for the converse in all unitary minimal $W$-algebras.