Lipschitz characterization of Fock spaces via the Fock distance

Let $F^p_\alpha$ be the Fock space on ${\mathbb C}^n$, let $L^p({\mathbb C}^n,dv)$ be Lebesgue $L^p$ space, and let $d_\alpha(z,w)$ be the distance function associated with the Fock space parameter $\alpha$. Suppose $\alpha>0$, $0<p\le\infty$, and $f$ is an entire function on ${\mathbb C}^n$. Fock-space Lipschitz conjecture. The function $f$ belongs to $F^p_\alpha$ if and only if there exists a non-negative continuous function $g\in L^p({\mathbb C}^n,dv)$ such that

$$|f(z)-f(w)|\le d_\alpha(z,w)\left[g(z)+g(w)\right]$$

for all $z,w\in{\mathbb C}^n$. This would provide a Lipschitz-type characterization of Fock spaces analogous to the known characterization of Bergman spaces; the paper notes that the relevant distance estimates are not yet sufficiently understood.