Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}_{l}$ and ${\mathbb{Q}}{l'}$-adic etale cohomology for two (distinct) primes $l,l'\neq p$ (i.e. the Standard Conjecture D is fulfilled). Then it was proved in: Smirnov O., Graded associative algebras and Grothendieck standard conjectures, Invent. Math., 128 (1997), 201-206 that the Lefschetz standard conjecture holds also; hence the Kunneth decompositions of the motif of a smooth projective $P$ exists both with respect to $\mathbb{Q}_l$-adic and with respect to $\mathbb{Q}{l'}$-adic etale cohomology. Are these two decompositions necessarily isomorphic? I suspect that that the answer is 'Yes' and the proof is easy, but I am not sure.

P.S. I don't understand why writing ${\mathbb{Q}}_{l'}$ in my question leads to catostrophic appearance.

Suppose that over some (algebraically closed) field $K$ of characteristic $p>0$ we have: numerical equivalence of cycles coincides with homological one with respect to ${\mathbb{Q}}_{l}$ and ${\mathbb{Q}}_{l'}$-adic etale cohomology for two (distinct) primes $l,l'\neq p$ (i.e. the Standard Conjecture D is fulfilled). Then it was proved in: Smirnov O., Graded associative algebras and Grothendieck standard conjectures, Invent. Math., 128 (1997), 201-206 that the Lefschetz standard conjecture holds also; hence the Kunneth decompositions of the motif of a smooth projective $P$ exists both with respect to $\mathbb{Q}_l$-adic and with respect to $\mathbb{Q}_{l'}$-adic etale cohomology. Are these two decompositions necessarily isomorphic? I suspect that that the answer is 'Yes' and the proof is easy, but I am not sure.

P.S. I don't understand why writing ${\mathbb{Q}}_{l'}$ in my question leads to catostrophic appearance.