Unitarity conjecture for the exceptional vertex operator algebras U_6 and U_7

Let

$$U_6=\mathscr{W}_{-h^\vee+15/11}(E_6,f_{\mathrm{subreg}}),\qquad U_7=\mathscr{W}_{-h^\vee+21/16}(E_7,f_{\mathrm{subreg}}).$$

The source observes that both vertex algebras are positive, meaning that every irreducible module other than the algebra itself has positive conformal dimension. Unitarity conjecture. The vertex operator algebras $U_6$ and $U_7$ are unitary. Positivity motivates the expectation, but the supplied text gives no resolution.