Quantized six-vertex model at roots of unity: Frobenius property and free-parafermion spectra

We study the quantized six-vertex (Q6V) model on arbitrary admissible diagrams at $q=\varepsilon$, where $\varepsilon$ is a primitive $N$th root of unity with $N$ odd. For the commuting layer transfer matrices $\mathbb{T}(y)$ with mixed boundary conditions, we establish the quantum Frobenius propert