Maximal integral packing conjecture for product tori on the blown-up projective plane

For each integer $m\geq 2$, let $(Y_m,\omega_m)$ be the toric symplectic manifold obtained from $\mathbb{CP}^{2}$ with lines of area $m+2$ by symplectically blowing up along the standard Darboux ball of capacity $m$ centered at $[1:0:0]$. Let $L_{k,l}$ denote the product torus associated with the integer point $(k,l)$ in the moment image, where $k+l=m+1$. Maximal integral packing conjecture. The collection

$$\{L_{k,l}:k+l=m+1\}$$

is a maximal integral packing of $(Y_m,\omega_m)$ for each $m\geq 2$. This conjecture concerns the non-monotone toric manifolds $(Y_m,\omega_m)$, where the moment image contains $m$ relevant integer points and the corresponding product tori give a natural candidate packing. The claim is presented as an open problem; the paper's methods do not directly establish it because ambient monotonicity is lost.