[The  Shub conjecture](https://shub.ccny.cuny.edu/articles/Entropy%20Conjecture%20Conversation%20warwick.pdf) 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.