Riemannian inequality conjecture for positive total Gauss curvature

Let $\varSigma$ be a complete surface in $\mathbb R^3$ satisfying the stated hypotheses, and let $\kappa$ be its Gauss curvature. Assume $\kappa$ satisfies the stated integrability condition and that the total Gauss curvature is positive:

$$\int_\varSigma\kappa>0.$$

Let $H$ be the mean curvature and let $B(R)$ denote the relevant geodesic ball. Positive-curvature Riemannian conjecture. There exists $\varepsilon_0>0$ such that, for every $R>0$, there is a function $j\in\mathcal C_0^\infty(\varSigma\setminus B(R))$ satisfying the inequality

$$\left(\int_\varSigma Hj\,d\varSigma\right)^2>\varepsilon_0\int_\varSigma\left(|\nabla j|^2+j^2\right)d\varSigma.$$

This purely Riemannian statement would imply the thin quantum-layer conjecture under the hypotheses above. The paper does not prove it and presents it as an open geometric problem.