Bernard's homotopy-transfer conjecture for the higher-genus KZB equation

Let $g>1$. The universal version of the KZB equation is a holomorphic flat connection.

Homotopy-transfer conjecture. The connection can be constructed via the homotopy transfer theorem, using constructions (1) and (2).

This would extend the homotopy-transfer construction of the KZB connection from the punctured torus and its configuration spaces to higher-genus Riemann surfaces. The source gives no resolution of this conjecture.