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.

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.

Note that the book mentioned above uses a somewhat strange definition of fppf topology: fppf covers are required to be faithfully flat and of finite presentation (as opposed to faithfully flat and locally of finite presentation); in particular, some Zariski covers are not fppf. It seems to me that this leads to a lot of awkwardness (e.g., on p. 201). For the sake of definiteness though, let us say that for the purposes of this question fppf cohomology is computed when the covers are taken to be faithfully flat and locally of finite presentation, although I would appreciate if someone could comment on what difference this choice makes in comparison to what is used in the book.