Uniform Hessian bound for the fermionic free energy

Let $X$ be a complex manifold of complex dimension $n$, let $(\phi,\mu)$ be a weakly regular weighted measure on $X$, and let $u$ be a Lipschitz continuous function on $X$. For each positive integer $k$, let $\mathcal{F}_k$ denote the corresponding fermionic free energy.

Uniform Hessian bound. There is a constant $C$, independent of $k$, such that

$$0\leq-\left.\frac{d^{2}\mathcal{F}_{k}[k(\phi+tu)]}{dt^{2}}\right|_{t=0}\leq Ck^{n-1}.$$

This conjectural estimate asserts an order-$k^{n-1}$ bound for the second variation of the finite-$k$ free energy under a Lipschitz perturbation of the weight. The supplied text does not state whether the bound is proved or remains open.