The compactness and deformation-dimension conjecture for CFT moduli

Fix $c_0\geq 0$ and $E_{\min}>0$, and let ${\cal M}_{c\leq c_0}^{E_{\min}}$ be the moduli space of irreducible conformal field theories with central charge $c\leq c_0$ and

$$\min\{p+q>0\mid H^{p,q}\neq 0\}\geq E_{\min}.$$

For a given CFT, its minimal versal deformation is a deformation with the smallest possible base. Compactness and deformation-dimension conjecture. ${\cal M}_{c\leq c_0}^{E_{\min}}$ is a compact real analytic stack of finite local dimension, and the dimension of the base of the minimal versal deformation of any given CFT is at most $\dim H^{1,1}$. This conjecture proposes compactness of bounded-energy CFT moduli and a uniform local deformation bound; the source gives no resolution or evidence of a proof.