Let $K$ be a finite extension of the $p$adic numbers. Suppose that $V$ and $W$ are two (finite dimensional, $p$adic) continuous representations of $G_K$. Suppose that $V \otimes W$ is crystalline. Is $V$ crystalline up to twist by a character of $G_K$?

I'm indeed pretty sure that the answer is "yes". I'd prefer not to post the idea of the proof here because I asked one of my PhD students to write it down with all the details. 

