Song's refined-character formula for the Macdonald index

Let ${\cal V}$ be the strongly finitely generated vertex operator algebra associated with a four-dimensional ${\cal N}=2$ superconformal field theory, and let

$$Z_{\mathcal{V}}^{\mathsf{ref}}(\boldsymbol{z};q,t)=\sum_{n=0}^{\infty}\sum_{h\in\mathds{Z}}\mathsf{ch}({\cal V}^{(n)}_h;\boldsymbol{z})q^{h-n}t^n$$

be its refined character. Song's formula. The Macdonald index is the refined character:

$$\mathcal{I}_M(\boldsymbol{z};q,t)=Z_{\mathcal{V}}^{\mathsf{ref}}(\boldsymbol{z};q,t).$$

This identifies a protected four-dimensional index with filtered representation-theoretic data of the associated vertex operator algebra. The source presents it as an expected relation and gives no resolution status.