  We study eigenvalues along periodic cycles of post-critically finite
endomorphisms of $\mathbb{CP}^n$ in higher dimension. It is a classical result
when $n = 1$ that those values are either $0$ or of modulus strictly bigger
than $1$. It has been conjectured in [Van Tu Le. Periodic points of
post-critically algebraic holomorphic endomorphisms, Ergodic Theory and
Dynamical Systems, pages 1-33, 2020] that the same result holds for every $n
\geq 2$. In this article, we verify the conjecture for the class of weakly
post-critically finite all the way down maps which was introduced in [Matthieu
Astorg, Dynamics of post-critically finite maps in higher dimension, Ergodic
Theory and Dynamical Systems, 40(2):289-308, 2020]. This class contains a
well-known class of post-critically finite maps constructed in [Sarah Koch,
Teichm\"uller theory and critically finite endomorphisms, Advances in
Mathematics, 248:573-617, 2013]. As a consequence, we verify the conjecture for
Koch maps.
