The entropy inequality for real exponents

Let $H(x):=-x\log x-(1-x)\log(1-x)$ be the binary entropy. For real $k\geq 1$, let $\alpha_k\in(0,1)$ be the unique solution of

$$\alpha_k=\frac{1}{(1+\alpha_k)^{k-1}}.$$

The entropy inequality for real exponents. For $0\leq x\leq 1$,

$$\alpha_k H(x^k)\geq x^{k-1}H(x).$$

Equality holds at $x=0$, $x=\frac{1}{1+\alpha_k}$, and $x=1$. The inequality is known for some integer and fractional exponents, while the stated real-exponent version is presented as the paper's conjectural target.