The GUE corners limit conjecture in the diverging-temperature regime

Let Assumption $\mathrm{(function)}$ hold, and set $q=q(N)=e^{-\gamma(N)/N}$, where $\gamma(N)\to+\infty$ and $\gamma(N)\ll N$. Let $\lambda^r_j(N)$ denote the parts of the random lozenge-tiling signatures, and let $K$ be fixed. The condition $\mathsf{f}'(1)<0$ concerns the slope of the weakly decreasing function $\mathsf{f}$ at the left end of the top row. GUE corners limit conjecture. If $\mathsf{f}'(1)<0$, then an analogue of the main GUE-corners limit theorem holds under a suitable normalization determined by $\mathsf{u}(N)$ and $\upsigma^2(N)$, with

$$\lim_{N\to\infty}\mathsf{u}(N)=\lim_{N\to\infty}\upsigma^2(N)=0,$$

and

$$\left\{\frac{\lambda^r_j(N)-\mathsf{u}(N)N}{\upsigma(N)\sqrt{N}}:1\le r\le K,\ 1\le j\le r\right\}\to\left\{\mathsf{L}^r_j:1\le r\le K,\ 1\le j\le r\right\}\sim\mathsf{GUE}_{K\times K}(1).$$

The conjecture concerns fluctuations at a scale smaller than $\sqrt{N}$ when the parameter $q$ approaches $1$ with $\gamma(N)\to\infty$ and $\gamma(N)\ll N$. The supplied text does not state whether this analogue has been proved, so its status remains open.