Let $f\colon X \rightarrow S$ be a proper morphism of schemes. Is the cohomology group $H^2(S, f_* \mathbb{G}_m)$ the same regardless of whether it is computed in the etale or the fppf topology? And if so, why?

This is claimed in the middle of p. 203 of "Neron models" by Bosch, Lutkebohmert, and Raynaud, and it is hinted there that Stein factorization is of use for proving the claim (Stein factorization also has a non-Noetherian version, by the way, see http://stacks.math.columbia.edu/tag/03H2). I would be grateful if someone could spell out the argument.