Nonexistence conjecture for nonnegative weighted scalar curvature on Oeljeklaus–Toma manifolds

Let $X(K,U)$ be an Oeljeklaus–Toma manifold. For a pluriclosed metric, let $R^{H,\psi}$ denote the weighted scalar curvature determined by the associated generalized Ricci-flow data and dilaton potential. Nonexistence conjecture. The manifold $X(K,U)$ admits no pluriclosed metric satisfying

$$R^{H,\psi}\geq 0.$$

The question arises because nonnegativity of $R^{H,\psi}$ is preserved along the relevant flow, so the assertion would amount to an elliptic obstruction on Oeljeklaus–Toma manifolds. The excerpt does not provide a resolution.