Let $X$ be a Voevodsky's motif (over a perfect field) that belongs to the positive part of the homotopy $t$-structure (i.e. its cohomology as an object of $D^-(ShSmCor)$ is zero in negative degrees). Is the same true for $X(1)$?

This seems to be a difficult question, and I will try to express my ideas about it. The Beilinson-Soule conjecture predicts that the answer is positive for $X=\mathbb{Z}(j)$, $j\ge 0$. Moreover, for any geometric motif $X$ this conjecture together with Poincare duality yields that there exists a constant $c_X$ such that $X(i)[c_X]$ is $t$-positive for any $i\ge 0$.

Besides, one can certainly consider motives either with $\mathbb{Z}/l\mathbb{Z}$-coefficients or with $\mathbb{Q}$-ones in this question. For the first of this possibilities, one can replace the Beilinson-Soule conjecture with the Bloch-Kato one (so the statements above become unconditional). Yet the Bloch-Kato conjecture does not seem to answer my question.

I would be deeply grateful for any ideas or (counter)examples for my question.

More generaly, it would be interesting to understand how much negative cohomology can the tensor product of two $t$-positive motives have.