Boundedness of the harmonic Fock projection in arbitrary dimension

Let $n\geq 2$, let $d\mu_{\beta}$ be the Gaussian measure on $\mathbb{R}^{n}$ with parameter $\beta>0$, and let $P_{\alpha}$ denote the harmonic Fock-space projection for $\alpha>0$. For $0<p<\infty$, consider its action on $L^{p}(\mathbb{R}^{n},d\mu_{\beta})$. Harmonic Fock projection boundedness conjecture. The operator $P_{\alpha}$ is not bounded for $0<p<1$; for $p\geq 1$, it is bounded if and only if $p\beta=2\alpha$. The theorem proves this characterization when $n=2N$ is even, while necessity of $p\beta=2\alpha$ is established for every $n\geq 2$. Extending the sufficiency to arbitrary dimensions $n\geq 3$ remains open.