Burbea's coefficient inequality conjecture for Hardy spaces

Let $f(z)=\sum_{n=0}^{\infty}a_nz^n$ belong to $H^p$ on the unit disc, and for $\alpha\geq1$ define $c_\alpha(n)$ by

$$\frac{1}{(1-z)^\alpha}=\sum_{n=0}^{\infty}c_\alpha(n)z^n,\qquad c_\alpha(n):=\binom{n+\alpha-1}{n}.$$

Burbea's conjecture. The inequality

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

holds for every $0<p\leq2$. This extends Carleman's inequality and would provide the coefficient estimate underlying the paper's study of contractive inequalities for Hardy spaces; it is known when $p$ is the reciprocal of a positive integer, while the full range remains unresolved in the source.