Higher-dimensional moment asymptotics for volumes of Kostlan zero sets

Let $M$ be a real algebraic manifold of dimension $n$, let $s_d$ be a random section in the complex Fubini--Study model in dimension $n$ and codimension $r\in\{1,\dots,n\}$, and let $\nu_d$ denote the volume measure of the real zero set of $s_d$. For $p\geqslant 3$ and $\phi_1,\dots,\phi_p\in\mathcal{C}^0(M)$, define $m_p(\nu_d)(\phi_1,\dots,\phi_p)$ as the $p$-th joint moment functional. Higher-dimensional moment conjecture. As $d\to +\infty$,

$$m_p(\nu_d)(\phi_1,\dots,\phi_p)=d^{\frac{p}{2}(r-\frac{n}{2})}\sigma_{n,r}^p\sum_{\mathcal{I}\in\mathcal{P\!P}_p}\prod_{\{i,j\}\in\mathcal{I}}\left(\int_M\phi_i\phi_j\left\lvert\operatorname{d}\!V_M\right\rvert\right)+o\left(d^{\frac{p}{2}(r-\frac{n}{2})}\right).$$

In particular, for $\phi\in\mathcal{C}^0(M)$,

$$m_p(\langle\nu_d,\phi\rangle)=\mu_p d^{\frac{p}{2}(r-\frac{n}{2})}\sigma_{n,r}^p\left(\int_M\phi^2\left\lvert\operatorname{d}\!V_M\right\rvert\right)^{\frac{p}{2}}+o\left(d^{\frac{p}{2}(r-\frac{n}{2})}\right).$$

Here $\mathcal{P\!P}_p$ denotes the set of pair partitions of $\{1,\dots,p\}$, and $\mu_p$ is the corresponding number of pairings when $p$ is even, with the usual vanishing interpretation for odd $p$.