  Let $A$ be a non-isotrivial almost ordinary Abelian surface with possibly bad
reductions over a global function field of odd characteristic $p$. Suppose
$\Delta$ is an infinite set of positive integers, such that
$\left(\frac{m}{p}\right)=1$ for $\forall m\in \Delta$. If $A$ doesn't admit
any global real multiplication, we prove the existence of infinitely many
places modulo which the reduction of $A$ has endomorphism ring containing
$\mathbb{Z}[x]/(x^2-m)$ for some $m\in \Delta$. This generalizes the
$S$-integrality conjecture for elliptic curves over number fields, as proved in
arXiv:math/0509485, to the setting of Abelian surfaces over global function
fields. As a corollary, we show that there are infinitely many places modulo
which $A$ is not simple, generalizing the main result of arXiv:1812.11679 to
the non-ordinary case.
