I believe this should be a well known result, but I wasn’t able to prove or find a good reference for it.

Let $E$ and $F$ be $n$-regular, respectively $m$-regular vector bundles in the sense of Castelnuovo-Mumford, the is it true that $E\otimes F$ is at most $n+m$-regular?

The result is true for modules, let’s say $M$ and $N$ provided that $Tor^1(M,N)$ is zero, due to a result by Caviglia. This is the case here locally, so I would like to claim it for the vector bundle case, but I am not sure if this holds, because of the difference in the definition of the regularity for modules and vector bundles.

Any help or reference will be appreciated.