The symmetry-group dimension conjecture for conformal field theories

Let $H=\bigoplus_{p,q}H^{p,q}$ be the state space of a conformal field theory, with $H^{1,0}$ its subspace of states of conformal weights $(1,0)$, and let the group of symmetries mean the automorphisms of $H$ preserving all the CFT structures. Symmetry-group dimension conjecture. The group of symmetries is a compact Lie group of dimension at most $\dim H^{1,0}$. This conjecture predicts a finiteness and dimensional bound for continuous symmetries of a CFT; the source gives no resolution or supporting result.