Hall--Littlewood law of large numbers with a positive Plancherel parameter

Let $\lambda(n)$ be a random Young diagram under the Hall--Littlewood coherent measures indexed by a triplet $(\boldsymbol\alpha;\boldsymbol\beta;\gamma)$ satisfying

$$\boldsymbol\alpha=(\alpha_1\ge\alpha_2\ge\cdots\ge0),\qquad \boldsymbol\beta=(\beta_1\ge\beta_2\ge\cdots\ge0),\qquad \gamma\ge0,$$

and

$$\sum_{i=1}^{\infty}\alpha_i+\sum_{i=1}^{\infty}\frac{\beta_i}{1-\mathfrak q^{-1}}+\frac{\gamma}{1-\mathfrak q^{-1}}=1.$$

Positive-Plancherel-parameter conjecture. The technical assumption $\gamma=0$ in the law of large numbers can be dropped: the same convergence should hold for every triplet $(\boldsymbol\alpha;\boldsymbol\beta;\gamma)$ satisfying the displayed conditions.