Fonarev's conjecture on the mutation Lefschetz basis for Grassmannians

Let ${\mathsf{Gr}}(k,V)$ be a Grassmannian, let $\mathrm{Y}^{\mathrm{mu}}_{k,n-k}$ denote the lexicographically minimal upper-triangular representatives of the cyclic orbits of Young diagrams, let $\Sigma^\lambda\mathcal{U}^*$ be the corresponding Schur functor of the dual tautological bundle, and let $o(\lambda)$ be the length of the cyclic orbit containing $\lambda$. Fonarev's conjecture. The orthogonal $\mathcal{R}^{\rm mu}$ in the stated semiorthogonal decomposition vanishes; equivalently, $\langle \Sigma^\lambda\mathcal{U}^*\mid\lambda\in\mathrm{Y}^{\mathrm{mu}}_{k,n-k}\rangle$ is a Lefschetz basis with support function $o$. This conjecture concerns the expected fullness of the mutation construction for Grassmannians; the source records the vanishing as expected, and no resolution is supplied here.