Burgers-equation conjecture for character-measure limit shapes

Let $\mu_{n,k}(\lambda\mid\{x_i\}_{i=1}^n,\{y_j\}_{j=1}^k)$ be the character measure on partitions $\lambda\subseteq k^n$ defined using Schur functions, and take $x_i=e^{\varphi(i/n)}$ and $y_j=e^{\psi(j/n)}$ for smooth functions $\varphi$ and $\psi$. Character-measure limit-shape conjecture. As $n,k\to\infty$, the limit shape of this character measure is described by the Burgers equation. The proposed description connects asymptotic character measures for skew Howe dual pairs with the Burgers-equation asymptotics of the Harish-Chandra--Itzykson--Zuber integral, but the source does not state a resolution.