Height scaling conjecture for infinite-variance simply generated trees

Let ${\mathcal T}_n$ be the conditioned simply generated tree, and let $H({\mathcal T}_n)$ denote its height. Assume that the condensation parameter satisfies $\nu=1$ and that the offspring variance is infinite, $\sigma^2=\infty$. Height scaling conjecture.

$$\frac{H({\mathcal T}_n)}{\sqrt n}\overset{\mathrm p}{\longrightarrow}0.$$

For finite variance, height is of order $\sqrt n$ with a Brownian-excursion limit; the source conjectures this degenerate scaling in the boundary infinite-variance case, and gives no proof.