The entropy polynomial root conjecture

Let $k>r\geq 1$ be integers. Define the entropy polynomial

$$h_{k,r}(x):=\sum_{j=0}^{k-1}x^{rj}\sum_{v=0}^j\frac{(-1)^{j-v}}{v+1}\binom{rv+k}{k}\binom{k}{j-v}.$$

Let $\alpha_k$ satisfy $\alpha_k=(1+\alpha_k)^{1-k}$, and define

$$p_{k,r}(x):=\alpha_{k/r}k(1-x^r)^k h_{k,k}(x)-r(1-x^k)^k h_{k,r}(x).$$

The entropy polynomial root conjecture. The polynomial $p_{k,r}(x)$ has exactly two real roots in $(0,1)$, counting multiplicity. The paper explains that this root statement would imply the entropy inequality and would yield proofs for rational exponents through finite calculations, but it remains unproved in the stated generality.