I have been reading Bhatt's notes on perfectoid spaces and I have stumbled upon a fact whose proof I am unable to understand. Specifically, in Remark 4.2.8 Bhatt describes the functor $A\mapsto A_{!!}$ from $R^a$-algebras (almost $R$-algebras) to $R$-algebras which is left adjoint to the almostification functor (which is just the localization functor to the category of almost algebras). Bhatt gives as an exercise the statement that even though $(-)_{!!}$ doesn't preserve flatness, it does preserve faithful flatness.

I have been unable to solve this myself, so I followed the reference to Gabber-Ramero's Almost ring theory, Remark 3.1.3. Their argument is short so I replicate it here:

Let $\phi:A\to B$ be a morphism of almost algebras. Then $\phi$ is a monomorphism iff $\phi_{!!}$ is injective; moreover, $B_{!!}/\operatorname{Im}(A_{!!})\cong B_!/A_!$ is flat over $A_{!!}$ if and only if $B/A$ is flat over $A$, by proposition 2.4.35.

I am able to follow the claims in this argument, but I don't see how it implies faithful flatness. I suspect there is some faithful flatness criterion at play which I am not aware of, but I couldn't find it in the literature I've checked. Could someone explain how exactly one recovers the result from this argument?

Commutative AlgebraI, §3, no. 5, Proposition 9. $\endgroup$