Muckenhoupt-weight conjecture for extremal functions

Let $1<p<\infty$, let $\varphi_p$ be the extremal function, and let $\mathscr{Z}(\varphi_p)$ be its zero set. Write $\operatorname{dist}(x,\mathscr{Z}(\varphi_p)\cup\{0\})$ for the distance from $x$ to that set. Muckenhoupt-weight conjecture. The function

$$\left(\frac{|x\varphi_p(x)|}{\operatorname{dist}(x,\mathscr{Z}(\varphi_p)\cup\{0\})}\right)^p$$

is a Muckenhoupt $A_p$ weight.