Hadamard convergence conjecture for smooth hypersurfaces in projective space

Let $V_d:=\Gamma(\mathbb{P}^n,\mathcal{O}(d))$ and let $U_d\subset V_d$ be the space of smooth hypersurfaces of degree $d$ in $\mathbb{P}^n$. Projective-space Hadamard convergence conjecture. In the Hadamard topology,

$$\lim_{d\to\infty}Z_{U_d/\mathbb{F}_q}\left(q^{-\dim V_d}t\right)=Z_{\operatorname{GL}_{n+1}/\mathbb{F}_q}\left(q^{-(n+1)^2}t\right).$$

In particular,

$$\frac{Z_{U_d/\mathbb{F}_q}(q^{-\dim V_d}t)}{Z_{\operatorname{GL}_{n+1}/\mathbb{F}_q}(q^{-(n+1)^2}t)}$$

converges uniformly on compact subsets of $\mathbb{C}$ to the constant function $1$. This is the explicitly separated $X=\mathbb{P}^n$ specialization of the proposed Hadamard Bertini refinement; the source gives no resolution.