Liouville theorem in the missing exponent region

Establish a Liouville theorem for the $(p,q)$-Laplacian inequality $\Delta_pu+\Delta_qu+u^s|\nabla u|^t\leq 0$ for all exponent pairs $(s,t)\in\mathbb{R}^2$, including the region $\{(s,t):q-1\leq s+t\leq p-1\}$ that is not covered by the stated results.

Background

The paper proves Liouville-type nonexistence results for positive weak solutions of the inequality $\Delta_pu+\Delta_qu+u^s|\nabla u|^t\leq 0$ under several parameter-region and volume-growth assumptions. In Euclidean space, Corollary 1.6 gives nonexistence conditions covering the regions $G_1$ through $G_4$.

The authors explain that their proof depends heavily on Lemma 2.1 and therefore does not cover every exponent pair. They identify $\{(s,t):q-1\leq s+t\leq p-1\}$ as an explicit example of a missing region, making extension of the Liouville theorem to this region an unresolved problem.