The height-function conjecture for bounded lecture hall tableaux

For a sequence of bounded lecture hall tableaux with limiting height function $h(x,y)=\lim h_n(x,y)/n$, let $u(x,y)$ be the solution of the Burgers equation introduced for the associated dimer model. Height-function conjecture. The function $u(x,y)$ satisfies

$$\operatorname{Im}(u)=\pi\frac{\partial h}{\partial y}$$

and

$$\arg(u)=\pi\left(\frac{\partial h}{\partial x}+1\right)$$

for some branch of the argument. This conjecture connects the complex Burgers-equation solution with the limiting height function, extending the correspondence known for periodic dimer models; the paper supports it with structural similarities and strong numerical evidence, but does not provide a proof.