Hall--Littlewood central limit conjecture for row and column lengths

Let $\lambda(n)$ be a random Young diagram with row lengths $\lambda_i(n)$ and column lengths $\lambda_i'(n)$. Assume that the positive $\alpha$-parameters and positive $\beta$-parameters are distinct, so that $\alpha_1>\alpha_2>\cdots$ and $\beta_1>\beta_2>\cdots$. Let the limiting means be those given by the law of large numbers, and normalize the row and column lengths by $\sqrt n$.

Central limit conjecture. The infinite vector

$$\{\lambda_1(n),\lambda_2(n),\ldots;\lambda_1'(n),\lambda_2'(n),\ldots\}$$

is asymptotically jointly Gaussian after subtracting its limiting means and normalizing by $\sqrt n$. The limiting covariances are $\alpha_i\mathbf 1_{i=j}-\alpha_i\alpha_j$ between $\lambda_i$ and $\lambda_j$, $\frac{\beta_i}{1-\mathfrak q^{-1}}\mathbf 1_{i=j}-\frac{\beta_i\beta_j}{(1-\mathfrak q^{-1})^2}$ between $\lambda_i'$ and $\lambda_j'$, and $-\frac{\alpha_i\beta_j}{1-\mathfrak q^{-1}}$ between $\lambda_i$ and $\lambda_j'$.

The conjecture predicts Gaussian fluctuations around the law-of-large-numbers frequencies. The source notes that the analogous result is known for the uniform measure on $\mathbb U$ and for related measures when the Hall--Littlewood parameter is zero, while the stated general case is not proved there.