Tate cycles on some quaternionic Shimura varieties mod p

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.