Evans–Gannon's modular-data conjecture for odd-order near-group categories

Let $G$ be a finite abelian group of odd order, let $q$ be a quadratic form on $G$, and let $\mathcal{NG}(G,q,b,c)$ be the corresponding near-group fusion category. Let $E_j$ be the simple objects in Theorem. Evans–Gannon's conjecture. There exists a metric group $(G',q')$ of order $|G|+4$ such that the simple objects $E_j$ are indexed by $g\in G$ and $x\in G'\backslash\{e\}$, with $E_{g,x}=E_{g,x^{-1}}$ and

$$\theta_{E_{g,x}}=\langle g,g\rangle e^{2\pi i\partial q'(x)}.$$

Moreover, the modular data are given by the Kronecker product of the Weil representation for $(G,q)$ with modular data $(S',T')$ for a rank $|G|+3$ modular category:

$$S^{q,q'}=S^q\otimes S',\qquad T^{q,q'}=T^q\otimes T'.$$

Here

$$T'=\operatorname{Diag}(1,1,\langle g,g\rangle_q,\langle x,x\rangle_{q'})_{g\in G,\,x\in G'}.$$

This conjecture predicts a uniform description of the modular data of the centers of odd-order near-group fusion categories by means of metric groups and Weil representations; the supplied text does not indicate whether it has been resolved.