Uniform torsion-freeness conjecture for smooth projective varieties

Let $X\subset\mathbb{P}^{N}$ be a smooth, projective variety of dimension $n$ and degree $D$ over an algebraically closed field $k$. For a prime number $\ell$ coprime to the characteristic of $k$, consider the torsion subgroup $\mathrm{H}^{i}(X,\mathbb{Z}_{\ell})_{\mathrm{tors}}$ of its $\ell$-adic cohomology.

Uniform torsion-freeness conjecture. There exist polynomials $\psi(x),\phi(x)\in\mathbb{Z}[x]$ such that

$$\mathrm{H}^{i}(X,\mathbb{Z}_{\ell})_{\mathrm{tors}}=0$$

for $0\leq i\leq 2n$ whenever

$$\ell>\psi\bigl(D^{\phi(N)}\bigr)$$

is a prime number coprime to the characteristic of $k$.

This conjecture proposes a uniform bound, polynomial in $D$ and controlled by $N$, beyond which the integral $\ell$-adic cohomology is torsion-free. It is motivated by bounds on Betti numbers and by Gabber's torsion-freeness theorem, which gives torsion-freeness for all but finitely many $\ell$ in positive characteristic; the stated uniform bound over arbitrary algebraically closed fields remains open.