Let $X/k$ be a smooth projective variety over a finite field of characteristic $p$ and $\mathcal{A}/X$ be an Abelian scheme.

Is then $H^1_\mathrm{SYN}(X,\mathcal{A}[p]) = H^1_\mathrm{fppf}(X,\mathcal{A}[p])$ finite?

This is true if $X$ is a curve, see [Milne, Arithmetic Duality Theorems http://jmilne.org/math/Books/ADTnot.pdf ], p. 292, Lemma III.8.9.