Dihedral commutant conjecture for dual-unitary qubit circuits

Let $\mathcal M'_T$ be the commutant of the algebra generated by the operators $M_a$ and $M_{ab,\iota}+R_{2t}M_{ab,\iota}R_{2t}$ on the periodic chain of $2t$ spins, where $R_{2t}$ is reflection of the time lattice and $\Pi_{2t}$ denotes translation on that chain. Dihedral commutant conjecture. The commutant $\mathcal M'_T$ is the linear span of the representation of the dihedral group $D_t$ on the periodic chain of $2t$ spins:

$$\mathcal M'_T=\operatorname{span}\{R_{2t}^n\Pi_{2t}^{2\tau}:\ \tau=0,1,\ldots,t-1,\ n=0,1\}.$$

The statement is presented without proof as the analogue of a preceding commutant characterization and would determine the infinite-system spectral form factor through $\dim\mathcal M'_T$.