Higher-dimensional subharmonic gradient inequality

Let $\Omega\subset\mathbb{R}^d$ be an open subset with smooth boundary. For $\lambda\in C^\infty(\overline{\Omega})$ with $\lambda>0$ on $\overline{\Omega}$, and $\phi,\psi\in C^2(\overline{\Omega})$ vanishing on $\partial\Omega$ and both subharmonic with respect to $-\nabla(\lambda\nabla)$, the higher-dimensional gradient inequality conjecture. There exists a constant $C_{d,\Omega}$ depending only on $d$ and $\Omega$ such that

$$\int_{\Omega}\lambda(x)|\nabla\phi(x)\mathbin{.}\nabla\psi(x)|\,dx\leq C_{d,\Omega}\int_{\Omega}\lambda(x)\nabla\phi(x)\mathbin{.}\nabla\psi(x)\,dx.$$

The source proves the corresponding one-dimensional inequality with optimal constant $3$ and proposes this extension to higher dimensions; no resolution is given in the supplied text.