The rigid Lyapunov–Harder-Narasimhan conjecture

Let $C$ be a Teichmüller curve of genus $g$, with Lyapunov exponents $\lambda_1\geq\cdots\geq\lambda_g$ and normalized Harder–Narasimhan numbers $w_1\geq\cdots\geq w_g$. Let $k$ be an index such that $1\leq k<g$, and let $\operatorname{VHS}$ denote the variation of Hodge structure associated with the family. Rigid Lyapunov–Harder-Narasimhan conjecture. If

$$\sum_{j=1}^{k}\lambda_j=\sum_{j=1}^{k}w_j$$

and $w_k\neq w_{k+1}$, then the variation of Hodge structure contains a local system of rank $2(g-k)$. The paper proposes this as a rigidity statement in the equality case of the main polygon conjecture.