The general Hadamard convergence conjecture for smooth sections

Let $X/\mathbb{F}_q$ be a smooth projective variety equipped with an ample line bundle $\mathcal{L}$, and let $U_d \subset \Gamma(X,\mathcal{L}^d)$ be the open subvariety of smooth sections. The Hadamard convergence conjecture states that, in the Hadamard topology on $\mathcal{H}_1$,

$$\lim_{d\rightarrow \infty} Z_{U_d}\left(q^{-\dim U_d}t\right)=\left(Z_X^{\mathrm{Kap}}\left([q^{-(\dim X+1)}]\right)\right)^{-1}$$

and

$$\left(Z_X^{\mathrm{Kap}}\left([q^{-(\dim X+1)}]\right)\right)^{-1}=\left(\prod_{k=1}^{\infty}Z_{\operatorname{Sym}^k X}\left(q^{-k(\dim X+1)}t\right)\right)^{-1}.$$

This conjecture strengthens the preceding convergence results, which hold separately in the Witt and weight topologies, by predicting convergence in their common Hadamard refinement. Its general validity remains open.