The Shub conjecture on topological entropy $h(f)$ of self map f on manifold M says that the topological entropy is greater (or equal) than (to) the log of maximum absolute values of the eigenvalues of the linear map $f_*$ induced on Homologies.

Are there some Polynomial entropy version of this conjecture on compact topological manifolds or even topological space?

The polynomial entropy is described here

https://link.springer.com/article/10.1134/S156035472304007X

I asked the question in Physicsoverflow and in a comment form in RG too.