Geometric decomposition conjecture for the K3 elliptic genus

Let $X$ be a K3 surface with holomorphic tangent bundle $T:=T^{1,0}X$, and let $\mathbb E_{q,-y}$ be the formal power series of holomorphic vector bundles used to define the geometric elliptic genus. Let $e(\tau)$ be the function in the K3 elliptic-genus decomposition. For each $n\in\mathbb N_{>0}$, let $p_n$ be a polynomial and write

$$p_n(T)=\sum_{j=0}^{N_n}a_jT^{\otimes j}\quad\text{if}\quad p_n(x)=\sum_{j=0}^{N_n}a_jx^j,$$

with $T^{\otimes 0}=\mathcal O_X$. Geometric decomposition conjecture. There are such polynomials $p_n$ satisfying

$$\mathbb E_{q,-y}=-\mathcal O_X\cdot\chi_0(\tau,z)+T\cdot\chi_{\rm mm}(\tau,z)+\sum_{n=1}^{\infty}p_n(T)\cdot q^n\widehat\chi(\tau,z),$$

and

$$e(\tau)=\sum_{n=1}^{\infty}\left(\int_X\operatorname{Td}(X)p_n(T)\right)\cdot q^n.$$

This conjecture seeks a geometric counterpart to the representation-theoretic decomposition of the K3 elliptic genus and is proposed as a step toward understanding Mathieu moonshine; the source gives no resolution.