This post is a follow up to the counterexample presented in the following questions If $Rm$ is free, how do you show $m \otimes n \neq 0$?. The hope now is that we can eliminate the pathological behavior presented in the counterexamples by imposing the condition that one of the modules we are working with be free. In particular I hope the following question formulated below is sufficient to avoid the nice counter arguments cited above.
Let $R$ be a commutative ring with identity. Let $M, N$ be $R$-modules with $m \in M$ and $n\in N$ with $n \neq 0$.
If $Rm$ is free and $N$ is free is $ m \otimes n \neq 0$?