I am trying to prove the following Weak-Lefschetz-type statement for motivic cohomology: if $Z$ is a smooth hyperplane section of a smooth projective $X$ of dimension $d$ (say, over the field of complex numbers), then for any integral $r\ge 0$ and $l>1$ the homomorphism $H^i(X,\mathbb{Z}/l\mathbb{Z}(r))\to H^i(Z,\mathbb{Z}/l\mathbb{Z}(r))$ is injective for $i=d-1$ and is bijective for $i\le d-2$. In particular, $Chow^j(X, \mathbb{Z}/l\mathbb{Z})$ should be isomorphic to $Chow^j(Z, \mathbb{Z}/l\mathbb{Z})$ if $2j\le d-2$.

Could this statement be true? Are there any counterexamples known, or is there any evidence supporting 'my' conjecture? I would be interested in any comments! Note that for motivic cohomology with rational coefficients the analogue of 'my' conjecture easily follows from the (conjectural!) existence of a 'reasonable' motivic $t$-structure (for Voevodsky's motives). Yet motivic conjectures tell almost nothing on motives and cohomology with torsion coefficients. Still, if they are true, it would be natural to conjecture that there exists a bound on the exponent of the kernel and the cokernel of the homomorphism $H^i(X,\mathbb{Z}(r))\to H^i(Z,\mathbb{Z}(r))$ for $i\le d-2$ (and this bound only depends on $d$?). Could this be true?

More generally, I would certainly like to understand which motivic conjectures become wrong for integral (and torsion coefficients), and 'how much' they are wrong.