Geometric interpretation of the differential on antisymmetrized moduli-space forms

Let $\mathsf{HGK}_0$ be identified with some version of compactly supported differential forms on moduli spaces of curves with antisymmetrized marked points. For the forgetful map

$$\pi:{\rM}_{g,n+1}\to {\rM}_{g,n},$$

let pullback along $\pi$ act on the corresponding compactly supported differential forms.

Geometric interpretation conjecture. Under this identification, the operation $\Delta$ on $\mathsf{HGK}_0$ corresponds to pullback under $\pi$, and this pullback induces a well-defined operation on compactly supported differential forms.

The statement is explicitly presented as a conjectural geometric interpretation, and the source gives no resolution status. The notation is only partially specified in the source.