The twisted $E_8$-equivariant lift conjecture for $e_8$

Let $e_8 e \pi_{-16}\operatorname{TMF}$ be the element specified by its image under $\Phi$, and let $\tau\colon BE_8\to K(\mathbb{Z},4)$ represent the generator of $H^4(BE_8;\mathbb{Z})\simeq\mathbb{Z}$. A Borel $E_8$-equivariant twisted lift should exist:

The twisted $E_8$-equivariant lift conjecture. The element $e_8$ lifts to

$$\hat e_8\in \operatorname{TMF}^{16+\tau}(BE_8).$$

Here the twist is obtained from the canonical map $K(\mathbb{Z},4)\to BO\langle 0,\ldots,4\rangle\to BGL_1(\operatorname{TMF})$. This supplies the twisted equivariant class needed for the subsequent power-operation and differential-geometric computations, but the existence of such lifts remains conjectural.