Divisible elements in the cohomology of Milnor $K$-theory

As a consequence of the strong Tate conjecture over finite fields one can deduce (see here proposition 8.20) for every smooth variety $X$ over a finite field:
$$H_{cont}^i(X,mathbb{Q}_l(n))=H_{Zar}^{i-n}(X,mathcal{K}_n^M)otimes mathbb{Q}_loplus H_{Zar}^{i-n-1}(X,mathcal{K}_n^M)otimes mathbb{Q}_l$$
Here $H_{cont}^i(X,mathbb{Q}_l(n))$ is the continuous etale cohomology with respect to the inverse system given by $mu_{l^k}^{otimes n}$ for different $k$s, tensored with rationals. The sheaf $mathcal{K}_n^M$ is the $n$-th Milnor $K$-theory sheaf.

A quick comparison implies that since $H_{cont}^i(X,mathbb{Z}_l(n))$ does not contain infinitely $l$-divisible elements (see here Cor 4.9), then any infinitely $l$-divisible element in $H_{Zar}^{i}(X,mathcal{K}_n^M)$ is torsion. I was wondering whether this is something one can prove (without the assumption of Tate’s conjecture)? Even for $n=1$ the Milnor $K$-theory sheaf seems to coincide with $mathbb{G}_m$, I’m not sure how to deduce the result.