Rosenthal compacta that are premetric of finite degree

We show that if a separable Rosenthal compactum $K$ is an $n$-to-one preimage of a metric compactum, but it is not an $n-1$-to-one preimage, then $K$ contains a closed subset homeomorphic to either the $n-$Split interval $S_n(I)$ or the Alexandroff $n-$plicate $D_n(2^\mathbb{N})$. This generalizes a result of the third author that corresponds to the case $n=2$.