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) thelog 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.