The QUP polar-code functional inequality assumption

Let $h:[0,1]\to\mathbb{R}$ be a non-negative function, and let $q>0$ be a constant such that

$$2h(z)=q\bigl(h(z^2)+h(2z-z^2)\bigr)$$

for every $z\in[0,1]$.

QUP polar-code functional inequality. Then

$$h(z_1)+h(z_2)\leq q\bigl(h(z_1z_2)+h(z_1+z_2-z_1z_2)\bigr)$$

for every $z_1,z_2\in[0,1]$.

This assumption is used in the analysis of QUP polar codes. The supplied text does not establish the implication or provide evidence resolving it, so its status remains open.