Poisson BF theory as the twist of pure 4d supergravity

Consider pure 4-dimensional $\mathcal N=1$ supergravity on $\mathbb R^4$. Let $\mathbb C^2$ carry its standard holomorphic symplectic structure, and write $\mathcal L_{2,0}=\Omega^{0,\bullet}(\mathbb C^2)$ for the local Lie algebra underlying Poisson BF theory. 4d supergravity twist conjecture. The twist of pure 4-dimensional $\mathcal N=1$ supergravity on $\mathbb R^4$ is equivalent to Poisson BF theory on $\mathbb C^2$, namely the cotangent theory to $\mathcal L_{2,0}$. This is the four-dimensional analogue of the proposed five-dimensional supergravity relationship and gives a conjectural supergravity interpretation of holomorphic Poisson BF theory.