Conjecture on hyperbolic density in real-polynomial isentropes

Consider the space of real polynomials of degree $d>2$ with all critical points real. An isentrope is the set of polynomials having a fixed topological entropy $h$, and a polynomial is hyperbolic when the iterates of all critical points converge to attracting periodic points and there are no neutral periodic points.

Hyperbolic-density conjecture. In the space of polynomials of degree $d>2$ with all critical points real, there are no isentropes of entropy

$$h\in(0,\log d)$$

where hyperbolic polynomials are dense.

This conjecture asserts a negative answer to Thurston's question about whether a dense set of entropy levels has hyperbolic polynomials dense in the corresponding isentropes. The source gives no resolution, so the conjecture remains open.