Conjecture on the unique singular point at infinity of GHV models

Let the GHV model be associated with a smooth hypersurface satisfying the assumptions of theorem, and let $n$, $d$, and $w$ be the parameters appearing there, with $n+d-w>0$. A compactification is a complete model containing the GHV model. The polar locus is the locus where the compactified model has poles, and a singular point at infinity is a singularity lying on the boundary of the compactification. The invariant $\nu_{P,0}$ denotes the local invariant at the point $P$ and value $0$. Unique-singularity conjecture. There exists a compactification for which the GHV models have only one singular point $P$ at infinity, located on the polar locus at finite distance, and for which

$$\nu=\nu_{P,0}=n+d-w.$$

The conjecture predicts a compactification with a single controlled singularity at infinity and specifies its local invariant. The supplied text does not state whether this prediction has been proved or disproved.