Macdonald law of large numbers for row and column lengths

Let $q,t\in[0,1)$ be Macdonald parameters, and let $\lambda(n)$ be a random Young diagram distributed according to the Macdonald measure $\mathscr M_n^{\boldsymbol\alpha;\boldsymbol\beta;\mathbf{Pl}_\gamma}$. Write $\lambda_i(n)$ for its row lengths and $\lambda_i'(n)$ for its column lengths.

Macdonald law-of-large-numbers conjecture. With almost sure convergence,

$$\frac{\lambda_i(n)}{n}\to\alpha_i, \qquad \frac{\lambda_i'(n)}{n}\to\beta_i\frac{1-q}{1-t}, \qquad i=1,2,\ldots.$$

This extends the Hall--Littlewood law of large numbers from one Macdonald specialization to the two-parameter Macdonald setting. The source presents it as a conjectural analogue of the proved Hall--Littlewood result.