Song's class formula conjecture for Lagrangian planes of $K3^{[n]}$-type

Let $X$ be a hyperkähler variety of $K3^{[n]}$-type, let $P\subset X$ be a Lagrangian plane, and let $\ell$ be the class of a line on $P$. Let

$$L=(\ell,-)\in H_2(X,\mathbb{Q})^{\vee}\cong H^2(X,\mathbb{Q})$$

be the class dual to $\ell$ with respect to the Beauville–Bogomolov–Fujiki form, so that $(L,x)=\int \ell\cup x$ for all $x\in H^2(X,\mathbb{Z})$. For a cohomology class $\gamma$, write $[\gamma]_k$ for its complex degree-$k$ component, and let $\operatorname{td}_X$ denote the Todd class of $X$.

Song's conjecture. The cohomology class of $P$ is

$$[P]=\left[\exp(L)\sqrt{\operatorname{td}_X}\right]_n.$$

This proposes a uniform extension, for every $n$, of formulas previously established in low dimensions for Lagrangian planes in varieties of $K3^{[n]}$-type. The source presents the formula as a conjecture and does not provide resolution evidence.