Universality conjecture for Gaussian fluctuations of random partitions

The parameters $\omega_n$ define models of random partitions, and $\omega_n$ converges to $0$ at speed $1/\sqrt{n}$ as $n$ tends to infinity. The fluctuations of the partitions are described by a Gaussian free field, up to a translation of the fluctuations on the $x$-axis. Universality conjecture. This asymptotic behaviour is universal among models of random partitions associated to parameters $(\omega_n)_{n\in\mathbb{N}}$ converging to $\mathbf{0}$ at speed $1/\sqrt{n}$ as $n$ goes to infinity. The conjecture proposes that the Gaussian-free-field fluctuation behaviour observed for Schur–Weyl measures extends universally to these random-partition models, independently of the specific model and, apart from translation, of the parameter.