Alternative q-analogue probability measures for partitions in a rectangle

Let $n$ and $k$ be positive integers, let $\lambda$ range over all partitions contained in an $n\times k$ rectangle, and write $\overline{\lambda}$ for the complementary partition and $\overline{\lambda}'$ for its conjugate. Let $\dim_q(V_{GL_n}(\lambda))$ and $\dim_q(V_{GL_k}(\overline{\lambda}'))$ denote the corresponding quantum dimensions. Alternative q-measure conjecture. The following expressions define probability measures on all such partitions:

$$\mu_{n,k}^{A2}(\lambda;q)=\frac{q^{\lVert\overline{\lambda}\rVert}\dim_q\bigl(V_{GL_n}(\lambda)\bigr)\,q^{\lVert\overline{\lambda}'\rVert}\dim_q\bigl(V_{GL_k}(\overline{\lambda}')\bigr)}{2\prod_{i=1}^{k+1}(q^i+1)^{k+2-i}\times\prod_{j=k+1}^n\prod_{i=1}^k(q^{j+2-i}+1)},$$ $$\mu_{n,k}^{A3}(\lambda;q)=\frac{q^{\lVert\overline{\lambda}\rVert}\dim_q\bigl(V_{GL_n}(\lambda)\bigr)\,q^{|\overline{\lambda}'|+\lVert\overline{\lambda}'\rVert}\dim_q\bigl(V_{GL_k}(\overline{\lambda}')\bigr)}{\prod_{i=1}^{2k}(q^i+1)^{k-|k-i|}\times\prod_{j=k+1}^n\prod_{i=1}^k(q^{j+k-i}+1)}.$$

These provide alternative $q$-analogues of the original measure, motivated by experimental data. The source presents them as conjectural and gives no resolution, so their normalization and probabilistic interpretation remain to be established.