Descent for Dualizable Modules

Question

It's known that a pure morphism of commutative rings $\phi:A\to B$ is of effective descent for the stack of modules. In other words if $\phi$ is pure one can recover $Mod(A)$ as the 2-limit of a truncated cosimplicial diagram of categories $Mod(B)\rightrightarrows Mod(B\otimes_AB)\Rrightarrow Mod(B\otimes_AB\otimes_AB)$. However, what if one replaces $Mod(-)$ with the stack of dualizable (finitely generated and projective) modules, $Mod^{fin}(-)$? I believe it is the case that dualizability descends along at least faithfully flat ring maps, which would imply, for instance, that faithfully flat maps of rings are of effective descent for dualizable modules. Presumably there are more maps which are of effective descent for $Mod^{fin}(-)$ than there are for $Mod(-)$. Is there anywhere that this question is addressed or solved?