Constancy of quantum walk spectral variety components and projection degrees

Let $d$ and $k$ be fixed, let ${\bf v}^{(1)},\ldots,{\bf v}^{(k)}$ be fixed vectors, and let $U$ vary over unitary matrices. Write ${\mathcal V}$ for the associated variety and let $\pi$ be the projection map on each component. Constancy conjecture. The number of components of ${\mathcal V}$ and the degrees of $\pi$ on each component are constant, except for a set of unitary matrices of positive codimension. This conjecture formalizes an empirical pattern observed in several dozen quantum random walks: these quantities appear to depend on the dimension and vector of chiralities, but not on the unitary matrix $U$, apart from an exceptional set.