The Quasi-diagonal Conjecture for invariant subvarieties

Fix $n>1$. For each $i\in\{1,\ldots,n\}$, let $\mathcal{S}_i$ be a component of a stratum of Abelian or quadratic differentials, and let $\mathcal{M}\subset\prod_i\mathcal{S}_i$ be a quasi-diagonal, meaning a prime invariant subvariety whose projection to every factor is dominant. A quasi-diagonal is trivial if either all factors are the same stratum and, at every point, all components agree up to rotation by $\pi$ and rescaling, or each pair of factors is either the same stratum or consists of a hyperelliptic component and its associated genus-zero stratum, with components agreeing up to rotation by $\pi$, rescaling, and quotienting by the hyperelliptic involution. The Quasi-diagonal Conjecture. All quasi-diagonals of rank at least two are trivial. The conjecture concerns the possible structure of invariant subvarieties with dominant projections and would rule out nontrivial higher-rank quasi-diagonals; the supplied text gives no resolution status.