Modified Lee–Zhang conjecture for symplectic and orthogonal Hitchin representations

Let $S$ be a connected closed oriented surface with fundamental group $\Gamma$. For a representation $\rho$, let $L_\rho$ denote its length spectrum for the specified invariant Finsler metric. The groups $\operatorname{PSp}(2k,\mathbb{R})$ and $\operatorname{PSO}(k,k+1)$ are viewed as subgroups of the corresponding projective linear groups.

Modified Lee–Zhang conjecture.

• For any Hitchin representation $\rho$ in $\operatorname{PSL}(2k,\mathbb{R})$, there is a Hitchin representation $j$ in $\operatorname{PSp}(2k,\mathbb{R})$ such that

$$L_j\leq L_\rho.$$

• For any Hitchin representation $\rho$ in $\operatorname{PSL}(2k+1,\mathbb{R})$, there is a Hitchin representation $j$ in $\operatorname{PSO}(k,k+1)$ such that

$$L_j\leq L_\rho.$$

This modifies the Lee–Zhang conjecture after the original statement was shown to fail for $n\geq4$. The source presents these two assertions as the proposed replacement; no resolution is supplied here.