Contractive coefficient inequality for Hardy spaces in the dual range

Let $f(z)=\sum_{n=0}^{\infty}a_nz^n$ be analytic in the unit disc, and for $\alpha\geq1$ let $c_\alpha(n)$ be defined by

$$\frac{1}{(1-z)^\alpha}=\sum_{n=0}^{\infty}c_\alpha(n)z^n.$$

Contractive dual coefficient conjecture. The inequality

$$\|f\|_{H^q}\leq\left(\sum_{n=0}^{\infty}|a_n|^2c_{q/2}(n)\right)^{1/2}$$

holds for every $2\leq q<\infty$. This is the symmetric companion to the coefficient inequality in the lower Hardy-space range; the source derives necessary conditions on the weights and presents the asserted contractivity as a conjecture, with no resolution supplied in the passage.