Hanusa–Nath positivity conjecture for affine type C_3^{(1)} core abaci

Let $\mathbb{N}$ denote the nonnegative integers, and let $\mathcal{C}_0$ be the set of core abaci of affine type $C_3^{(1)}$ with charge $0$. The preceding proposition identifies the values $|\lambda|$ for $\lambda\in\mathcal{C}_0$ with the values of the quadratic polynomial below. Hanusa–Nath's positivity conjecture. Each number in $\mathbb{N}-\{2, 12, 13, 73\}$ is of the form

$$6(k_1^2+k_2^2+k_3^2)+3k_1+k_2+5k_3,$$

where $k_1,k_2,k_3\in\mathbb{Z}$. This is an equivalent formulation of the positivity conjecture for affine type $C_3^{(1)}$, and it would show that the only excluded values in the set of sizes of charge-zero core abaci are $2,12,13$, and $73$.