Uniform positivity conjecture for correlation numbers along distinct cyclic Higgs rays

Let $X_0$ be a hyperbolic structure, and let $(\rho_t)_{t\geq 0}$ and $(\eta_t)_{t\geq 0}$ be rays associated to two different $d$-th order holomorphic differentials $q_1$ and $q_2$ on $X_0$. Assume that $q_1$ and $q_2$ have unit $L^2$-norm with respect to $X_0$ and that $q_1\neq -q_2$. Uniform positivity conjecture. The correlation number for the sum of the simple roots is uniformly bounded away from zero as $t$ tends to infinity:

$$M\left(\rho_t,\eta_t,\sum_{i=1}^{d-1}\alpha_i\right)$$

remains bounded below by a positive constant. This conjecture concerns the asymptotic correlation of renormalized Hilbert lengths along distinct rays in the Hitchin component and is motivated by the behavior established for cubic rays; whether the asserted uniform positive lower bound holds in general remains open.