Existence of constant-curvature ball packings under bounded vertex-degree variation

Let $\mathcal{T}$ be a triangulation with vertex set $V$, and let $d_i$ denote the vertex degree at $i\in V$. A real ball packing is an element of $\mathcal{M}_{\mathcal{T}}$, while a virtual ball packing is an element of $\mathds{R}_{>0}^N\setminus\mathcal{M}_{\mathcal{T}}$. A ball packing may therefore be real or virtual.

Constant-curvature packing conjecture. If

$$|d_i-d_j|\leq 10$$

for every $i,j\in V$, then there exists a real or virtual ball packing with constant curvature.

This conjecture proposes an existence criterion for constant-curvature packings without requiring the triangulation to be regular. The source gives no resolution or further evidence beyond the preceding convergence results, so its status remains open.