Conjecture on the Ising symmetry defect data

Let $\textbf{Ising}$ be the Ising fusion category, and consider its Deligne product $\textbf{Ising}\boxtimes\textbf{Ising}$ together with a $\mathds{Z}_2$-crossed extension containing a symmetry defect $X_1$. For $a\in\textbf{Ising}$, let $R^{X_1X_1}_{a\boxtimes a}$ denote the corresponding crossed braiding symbol, let $\theta_a$ be the twist of $a$, and let $[F^{X_1X_1X_1}_{X_1}]$ denote the indicated $F$-symbol matrix. Ising defect-data conjecture. There exists a set of solutions to the $\mathds{Z}_2$-crossed consistency equations associated to $\textbf{Ising}\boxtimes\textbf{Ising}$ such that

$$R^{X_1X_1}_{a\boxtimes a}=\theta_a \hspace{5pt} \text{ for } a\in\textbf{Ising}$$

and

$$[F^{X_1X_1X_1}_{X_1}]=S_{\textbf{Ising}}=\frac{1}{2}\begin{pmatrix}1&\sqrt{2}&1\\ \sqrt{2}&0&-\sqrt{2}\\ 1&-\sqrt{2}&1\end{pmatrix}.$$

This conjecture specifies algebraic data for the symmetry defect used in a proposed physical protocol for realizing a $T$-gate in topological quantum computing. A full solution of the $G$-crossed consistency equations is not known, and the stated existence remains open under the assumptions described in the source.