The unirationality conjecture for varieties with full étale-exceptional collections

Let $X$ be a variety over a field $k$. Assume that $X$ possesses a full étale-exceptional collection and has a $k$-point.

Unirationality conjecture. Then $X$ is $k$-unirational.

This is presented as a variant of Orlov's conjecture suggested by the paper's results. The supplied text gives examples motivating the statement but does not resolve it in general.