Higher-derivative local Fock-space contraction conjecture

Let $\mathcal{F}_\alpha^2$ be the Fock space, let $f^{(n)}$ denote the $n$th derivative of $f$, let $\mu_{n,\alpha}$ be the measure used to define the weighted derivative norm, and let $\Omega$ be a domain of finite measure $|\Omega|$. For an integer $n\geqslant 0$, the higher-derivative local contraction conjecture. For every $f\in\mathcal{F}_\alpha^2$,

$$\int_{\Omega} {|f^{(n)}(z)|^2}d\mu_{n,\alpha}(z) \le (1-e^{-(n+1)|\Omega|})\|f\|^2_{2,\alpha}.$$

This extends the known $n=0$ result of Nicola and Tilli to higher derivatives; the paper presents it as a conjecture and reduces it to further conjectures concerning the global derivative norm and Laguerre-polynomial coefficients.