Grothendieck limit-shape conjecture for extended parameter range

Let $\mathfrak{W}(u)=\mathfrak{W}(u\mid x,y,\beta)$ be the curve obtained from the surface and implicit-curve procedure described immediately beforehand, and consider Grothendieck random partitions with homogeneous parameters from the source. Extended Grothendieck limit-shape conjecture. The curve $\mathfrak{W}(u)$ is well-defined for every $\beta<\min(x^{-1},y)$, and it is the limit shape of the corresponding random partitions in the $45^\circ$-rotated coordinate system. This extends the proposed limit-shape description beyond the nonnegative-probability regime; the statement is presented as a conjecture, with no proof or resolution supplied in the source.