Existence of constant-curvature ball packings from equality of extended and ordinary Yamabe invariants

Let $\mathcal{T}$ be a triangulation. Write $\widetilde{Y}_{\mathcal{T}}$ for its extended combinatorial Yamabe invariant and $Y_{\mathcal{T}}$ for its ordinary combinatorial Yamabe invariant. A ball packing is either a real packing in $\mathcal{M}_{\mathcal{T}}$ or a virtual packing in $\mathds{R}_{>0}^N\setminus\mathcal{M}_{\mathcal{T}}$.

Yamabe-invariant equality conjecture. If

$$\widetilde{Y}_{\mathcal{T}}=Y_{\mathcal{T}},$$

then there exists a real or virtual ball packing with constant curvature.

The conjecture relates equality between the extended and ordinary combinatorial Yamabe invariants to the existence of a constant-curvature packing. The source gives no resolution or additional status evidence, so the conjecture remains open.