C^{1,1} geodesic conjecture for the coupled dHYM system

Let $(X,\alpha^{\mathbb{C}})$ be a complexified Kähler manifold in the hypercritical range, and let $\mathcal{H}$ denote the space of admissible potentials. C^{1,1} geodesic conjecture. If $(X,\alpha^{\mathbb{C}})$ is hypercritical, then for every $\varphi_0,\varphi_1\in\mathcal{H}$ there is a unique path $\varphi_t\in\mathcal{H}$ solving the coupled geodesic equation, joining $\varphi_0$ to $\varphi_1$, such that $\varphi_t\in\mathcal{C}^{1,1}(X,\mathbb{C})$. This is expected to extend the known $\mathcal{C}^{1,1}$ regularity for the degenerate dHYM geodesic equation to the coupled system, giving geodesic connectivity of the space of admissible potentials.