Existence conjecture for the moduli spaces of L-manifolds

A family of spaces (or, more precisely, Artin stacks) $\mathcal{L}_{0,n}$ should exist, equipped with the following structure. The spaces form a cyclic operad and have “forgetful” maps

$$p\colon \mathcal{L}_{0,n}\to\mathcal{L}_{0,m},\qquad m\leq n,$$

as well as “section” maps

$$x_i\colon \mathcal{L}_{0,n}\to\mathcal{L}_{0,n+1},\qquad 1\leq i\leq n+1,$$

which satisfy the obvious relations analogous to those for $\overline{\mathcal{M}}_{0,n}$. Moreover, the spaces should realize the previously defined spaces $H^\bullet_{\mathbf{n}}$ through isomorphisms

$$H^\bullet_{\mathbf{n}}\cong H^\bullet(\mathcal{L}_{0,n}).$$

This is one of the two questions proposed as a geometric analogue of the stable-curve/Frobenius-manifold picture for curved cyclic $\operatorname{Lie}_\infty$-algebras. The source presents it as a proposed existence statement, and the supplied parser gives no evidence that it has been resolved.