Let $X$ be a paracompact Hausdorff space and $R$ a commutative ring. All sheaves below will be sheaves of $R$-modules on $X$. A sheaf $S$ is soft if every section of $S$ over a closed subset can be extended over the whole space. Examples include the sheaf of continuous functions $X\to \mathbb{R}$ or, if $X$ is a smooth manifold, the sheaf of smooth functions $X\to\mathbb{R}$. Soft sheaves are useful because they are $\Gamma$-acyclic, so one can compute cohomology using soft resolutions.
Question: Is it true that if $S$ is a soft flat sheaf and $T$ an arbitrary sheaf of $R$-modules, then the product $S\otimes_R T$ is soft?
Comments: 1. The answer is yes if $R$ is a principal ideal domain (Bredon, Sheaf theory, 2nd edition, Corollary II.16.31). By inspecting the proof it is not too hard to generalize the latter statement to the case when (a) $X$ is arbitrary and $R$ has finite Tor dimension, or (b) $R$ is arbitrary and $X$ is finite dimensional (meaning there is an integer $d=\dim X$ such that $H^{>d}(X,S')$ vanishes for every sheaf $S'$). This suffices for many applications, but still it would be interesting to know whether these finiteness assumptions are really necessary.
2. If a sheaf $S$ is fine, i.e. if the endomorphism sheaf of $S$ is soft, then tensoring $S$ by an arbitrary sheaf results in a soft sheaf (regardless of whether or not $S$ is flat). For example, a soft sheaf of $R$-algebras is fine. But checking fineness is not always straightforward.
