  Let $F$ be a totally real field in which a prime number $p>2$ is inert. We
continue the study of the (generalized) Goren--Oort strata on quaternionic
Shimura varieties over finite extensions of $\mathbb F_p$. We prove that, when
the dimension of the quaternionic Shimura variety is even, the Tate conjecture
for the special fiber of the quaternionic Shimura variety holds for the
cuspidal $\pi$-isotypical component, as long as the two unramified Satake
parameters at $p$ are not differed by a root of unity.
