Moment convergence conjecture for generation sizes of simply generated trees

Let ${\mathcal T}_n$ be the conditioned simply generated tree, let $\widehat{{\mathcal T}}$ be its limiting size-biased tree, and let $l_k(T)$ denote the number of vertices at distance $k$ from the root. Generation-size moment conjecture. For every integer $r\geqslant1$ and every $k\geqslant1$, if $\nu>0$, then

$$\operatorname{\mathbb E}\,l_k({\mathcal T}_n)^r\to\operatorname{\mathbb E}\,l_k(\widehat{{\mathcal T}})^r\leqslant\infty.$$

The same convergence is conjectured for every real $r>0$, including non-integer moments. The source explains that integer moments would follow from uniform integrability, which is not established in the stated generality.