Weak-type level-set conjecture for normalized Hardy functions

Let $\mathbb{D}$ be the unit disc, let $\mu$ be the measure used in the paper on $\mathbb{D}$, and let $f\in H^2$. For $\lambda$ in the relevant range $0<\lambda<1$, consider the level set

$$\left\{z\in\mathbb{D}:|f(z)|^2(1-|z|^2)>\lambda\|f\|_{H^2}^2\right\}.$$

Weak-type level-set conjecture.

$$\mu\left(\left\{z\in\mathbb{D}:|f(z)|^2(1-|z|^2)>\lambda\|f\|_{H^2}^2\right\}\right)\leq\frac{1}{\lambda}-1.$$

The conjecture would imply the preceding Burbea conjecture through the distributional estimate discussed in the source. The supplied passage does not establish this sharp bound, so its resolution is left open here.