Tonkonog's cap class as the first Maurer–Cartan component

In Tonkonog's setup, let $\mathcal{BS}\in SH^0(X;\Bbbk)$ be the class obtained by counting holomorphic caps, and write $\beta=\beta_1+\beta_2+\cdots$ for the filtration components of the Maurer–Cartan element. Cap-class conjecture.

$$\mathcal{BS}=[\beta_1].$$

This identifies the leading Maurer–Cartan term with the cap-counting class whose closed–open image is the disc potential; the conjectural identification remains open in the generality stated.