Kac–Wakimoto principal-admissible character transformation conjecture

Let $\widehat{\mathfrak{g}}$ be an affine Lie superalgebra, let $\varphi:\widehat{\mathfrak{g}}\to\widehat{\mathfrak{g}}$ be a compatible homomorphism, and let $\Lambda\in\widehat{\mathfrak h}^*$ be a corresponding principal admissible weight. Define

$$\Lambda^0=\varphi^*(\Lambda+\widehat\rho)-\widehat\rho.$$

Let $\widehat R^ +$ and $\operatorname{ch}^ + _\Lambda$ denote the affine (super)denominators and normalized (super)characters. Kac–Wakimoto's principal-admissible conjecture. The normalized character identity

$$(\widehat R^ + \operatorname{ch}^ + _\Lambda)(h)=(\widehat R^ + \operatorname{ch}^ + _{\Lambda^0})(\varphi^{-1}(h)),\qquad h\in\widehat{\mathfrak h},$$

should hold. The source says that the analogous statement is proved for affine Lie algebras, while the Lie-superalgebra version is the conjectural extension; it also records that a related special case was proved in the cited work.