Equivalence classification conjecture for non-Abelian families

Let ${\cal C}$ be a topological order, and let ${\cal C}_{\boldsymbol a,K}$ and ${\cal C}_{\boldsymbol a',K'}$ be topological orders constructed from ${\cal C}$ using anyon data $\boldsymbol a,\boldsymbol a'$ and integer matrices $K,K'$. A trivial bilayer is a pair

$$\left(\begin{pmatrix} a\\ \mathbf{1}\end{pmatrix},\begin{pmatrix} -t_{a,a}&1\\1&0\end{pmatrix}\right),$$

where $a$ is an Abelian anyon in ${\cal C}_{Ab}$. The pairs $(\boldsymbol b,X)$ and $(\boldsymbol b',X')$ are direct sums of trivial bilayers. Equivalence classification conjecture. The topological orders ${\cal C}_{\boldsymbol a,K}$ and ${\cal C}_{\boldsymbol a',K'}$, with exactly the same chiral central charge rather than merely the same value modulo $8$, are equivalent if and only if, up to automorphisms of ${\cal C}$ and $GL(\mathbb Z)$ transformations,

$$(\boldsymbol a\oplus\boldsymbol b,K\oplus X)\sim(\boldsymbol a'\oplus\boldsymbol b',K'\oplus X').$$

The conjecture seeks a complete equivalence relation for the construction of non-Abelian families, extending the known equivalences from automorphisms of the starting topological order, unimodular integer transformations, and adjoining trivial bilayers. Its resolution would characterize when two such constructions yield the same topological order; the supplied text gives no evidence that the claim is proved or disproved.