Conjecture on constructing all inflations by truncation equalities

Fix $J\subseteq I$, and let $V_J$ be an object of $\mathcal{O}_{\nu}^{\lambda}(\mathbf R,\mathfrak g_J)$ for some $\lambda$, $\nu$, and $\mathbf R$. An inflation of $V_J$ is a construction of it as an object for $\mathfrak g$ extending its $\mathfrak g_J$-module structure. Inflation conjecture. Every inflation of $V_J$ to $\mathfrak g$ can be constructed using the equality of truncations in Lemma and the shift relation in. The conjecture concerns the classification of inflations from Levi-type subalgebras and is stated as intrinsically related to an earlier conjecture on such constructions; its resolution is not given here.