Levi negativity conjecture for the $L^1$-norm on quadratic differentials

Let $q_0\in\mathcal{Q}_g$ be generic, meaning that $q_0$ lies in the principal stratum. Let $T^H_{q_0}\mathcal{Q}_g$ denote a horizontal lift determined by a guiding frame, and let the Levi form be the complex Hessian of the $L^1$-norm function. Levi negativity conjecture. When $q_0\in\mathcal{Q}_g$ is generic, the Levi form of the $L^1$-norm function is negative on a horizontal lift $T^H_{q_0}\mathcal{Q}_g$. This complements the known positivity of the Levi form along the fiber directions and would describe the remaining horizontal sign pattern, with implications for the Levi convexity of unit ball bundles.