Weight conjecture for minimal nilpotent hypertoric characters

Let $N\geq 1$, and let $V(\Delta)$ be the boundary hypertoric vertex operator superalgebra for the minimal nilpotent orbit closure of $\mathfrak{sl}_N$. Weight conjecture. After removing the leading factor

$$\frac{\eta(q^2)^{2N}}{\eta(q)^{4N-2}},$$

the character is (quasi)modular of weight $2\lfloor N/2\rfloor$; consequently, the full character is (quasi)modular of weight $0$ if $N$ is odd and of weight $1$ if $n$ is even. The examples discussed in the paper support the claim, but the general assertion remains open.