Gauge-theory realization conjecture for Majorana, Dijkgraaf–Witten, and Gu–Wen anomalies

Let $k$ be as in the paper, let $x_1$ and $y$ specify the twist, and let $\alpha_{\mathrm{DW}}$ and $\alpha_{\mathrm{GW}}$ be the Dijkgraaf–Witten and Gu–Wen classes in $\mathit{SH}^5(B\mathbb{Z}/2\times B\mathbb{Z}/2^k,x_1,y)$. Gauge-theory realization conjecture. If

$$\beta\in\mathit{SH}^5(B\mathbb{Z}/2\times B\mathbb{Z}/2^k,x_1,y)$$

is in the subgroup spanned by $\alpha_{\mathrm{DW}}$ and $\alpha_{\mathrm{GW}}$, there is a (3+1)d $\mathbb{Z}/4\times\mathbb{Z}/2$ gauge theory with anomaly $\beta$ obtained by generalizing the constructions in the cited results. This is proposed because the required higher-unitary framework for time-reversal-symmetric SETs has not yet been systematically developed, so a complete rigorous construction remains open.