Character identification for the admissible affine VOA $V_{-3/2}(G_2)$

Let $V=V_{-3/2}(G_2)$ be the affine vertex operator algebra at admissible level $-3/2$, and let $f_0$ and $7f_{4/5}$ be the solutions of the fourth-order modular linear differential equation $(\flat_{-48/5})$ defined in the paper. The ordinary irreducible $V$-modules have conformal weights $0$ and $4/5$. Character identification conjecture. The characters of the irreducible ordinary modules over $V_{-3/2}(G_2)$ coincide with the solutions $f_0$ and $7f_{4/5}$ of $(\flat_{-48/5})$. If true, $f_0$ gives a character formula for $V_{-3/2}(G_2)$ in terms of modular forms. The conjecture concerns an admissible affine VOA that is not $C_2$-cofinite but is semisimple in the category $\mathcal{O}$; its status is not resolved in the supplied text.