Uniform boundedness conjecture for integral Tate obstructions in families

Let $S$ be a smooth, geometrically connected variety over $k$, with generic point $b7$, and let $f:X\rightarrow S$ be a smooth projective morphism. For each closed point $s\in |S|$, write

$$\widetilde{\mathrm{Ob}}_{\mathbb{Z}_\ell,s}=|((\widetilde{C}_{\mathbb{Z}_\ell,s})_{\mathrm{tors}})|.$$

For $d\geq 1$, let $|S|^{\leq d}$ be the set of closed points $s\in |S|$ with $[k(s):k]\leq d$, and define

$$\widetilde{\mathrm{Ob}}_{\mathbb{Z}_\ell}^{\leq d}:=\sup\{\widetilde{\mathrm{Ob}}_{\mathbb{Z}_\ell,s}\mid s\in |S|^{\leq d}\}.$$

Uniform boundedness conjecture. For every integer $d\geq 1$, one has

$$\widetilde{\mathrm{Ob}}_{\mathbb{Z}_\ell}^{\leq d}<+\infty$$

and

$$\widetilde{\mathrm{Ob}}_{\mathbb{Z}_\ell}^{\leq d}=1,\qquad \ell\gg 0.$$

This statement is predicted by the main conjecture of Bas, and asserts uniform boundedness, together with eventual vanishing of the torsion obstruction, across fibers of bounded residue degree.