Hikita probability lower-bound conjecture

Let $\mathbf{m}\in\mathbb{E}_n$, let $\lambda\vdash n$, and let $T\in\operatorname{HikSYT}(\mathbf{m},\lambda)$. For the recursively defined Hikita weight $h_T(q)$ and every real $\alpha\geq0$, Hikita probability lower-bound conjecture.

$$h_T(\alpha)\geq\frac{1}{\prod_i[\lambda_i]_{q=\alpha}!}.$$

The source explains that this pointwise estimate would imply a lower bound for the specialized elementary coefficient and is not established there.