Subgroup realization conjecture for low energy Ising field theories

Let $G$ be a finite group, let $\mathscr{M}_G$ denote the space of admissible functions on $G$ up to rescaling, and let $\Delta_G\subset\mathscr{M}_G$ be the locus of lattice systems at which phase transitions occur. Write $(\mathscr{M}_G\setminus\Delta_G)_{\operatorname{gapped}}$ for the subset representing gapped theories, and let $\mathrsfs{L}_{(G,\theta)}$ be the effective low energy topological field theory associated to $\theta$ in this subset. A subgroup $H\subset G$ determines a topological boundary theory $\mathrsfs{B}_{H}$. Subgroup realization conjecture. For every $\theta\in(\mathscr{M}_G\setminus\Delta_G)_{\operatorname{gapped}}$, the low energy field theory is $\mathrsfs{B}_{H}$ for some subgroup $H\subset G$. This proposes that low energy boundary theories arising from these gapped lattice systems have no nontrivial central extensions or decomposable boundary theories; the source gives no resolution of the claim.