The subharmonic gradient-product inequality

Let $B(0,1)\subset\mathbb{R}^d$ be the unit ball. Let $\lambda\in C^\infty(\overline{B(0,1)})$ satisfy $\lambda>0$ on $\overline{B(0,1)}$, and let $\phi,\psi\in C^2(\overline{B(0,1)})$ vanish on $\partial B(0,1)$ and be subharmonic with respect to the operator $-\nabla(\lambda\nabla)$. The subharmonic gradient-product inequality. There exists a constant $C_d$, depending only on the dimension $d$, such that

$$\int_{B(0,1)}\lambda(x)\lvert\nabla\phi(x)\cdot\nabla\psi(x)\rvert\,dx\leq C_d\int_{B(0,1)}\lambda(x)\nabla\phi(x)\cdot\nabla\psi(x)\,dx.$$

The paper states that this inequality would imply the difficult condition needed for its homogenization theorems and refers to earlier work and a monograph for context; its resolution is not established in the supplied text.