The universal framed KZB connection conjecture

Let $C$ be a smooth curve, let $\operatorname{Conf}(C,n)$ be the configuration space of $n$ distinct points on $C$, and let $\operatorname{Conf}^f(C,n)$ denote its framed version. Let $\hat{\mathfrak{t}}_{g,n}^f$ and $\hat{\mathfrak{t}}_{g,n}$ be the completed Lie algebras occurring in the corresponding principal bundles, and let $\lambda_i$ be the framing coordinate at the $i$-th point. The universal genus $g$ KZB connection is denoted by $\nabla_{g,n,\kappa}^{\operatorname{KZB}}$ with connection form $\alpha_{g,n}^{\operatorname{KZB}}$. The universal framed KZB connection conjecture. There is a flat universal framed KZB connection $\nabla_{g,n,\kappa}^{f\operatorname{KZB}}$ defined on the principal $\operatorname{exp}(\hat{\mathfrak{t}}_{g,n}^f)$-bundle over $\operatorname{Conf}^f(C,n)$ such that its pullback to the associated $\operatorname{exp}(\hat{\mathfrak{t}}_{g,n}^f)$-bundle over $C^n$ is

$$\nabla_{g,n,\kappa}^{f\operatorname{KZB}}:=\operatorname{d}-\alpha_{g,n}^{f\operatorname{KZB}},$$

where

$$\alpha_{g,n}^{f\operatorname{KZB}}:=\alpha_{g,n}^{\operatorname{KZB}}+\sum_{1\leqslant i\leqslant n}t_i\operatorname{d}\!\log(\lambda_i).$$

Moreover, the $1$-form $\alpha_{g,n}^{f\operatorname{KZB}}$ is $({\mathbb C}^\times)^n$-basic, and the induced connection on the $\operatorname{exp}(\hat{\mathfrak{t}}_{g,n})$-bundle over $\operatorname{Conf}(C,n)$ coincides with Enriquez's universal genus $g$ KZB connection. The conjecture proposes the framed extension needed to study the monodromy relations of higher-genus associators; its resolution status is not given in the source.