Asymptotic part-allocation conjecture for the B-$q$-Plancherel measure

Let $(\lambda^{(1)},\lambda^{(2)})$ be a random bipartition under the B-$q$-Plancherel measure, and let $\lambda=\lambda^{(1)}\sqcup\lambda^{(2)}$. Asymptotic part-allocation conjecture. The probability that the $(2k+1)$-st part of $\lambda$ belongs to $\lambda^{(1)}$ tends to

$$c_k=\frac{1}{2}\left(1+\frac{(q;q^{2})_{\infty}}{(q^{2};q^{2})_{\infty}}\,\frac{(q;q^{2})_{k}}{(q^{2};q^{2})_{k}}\,q^{k}\right).$$

The probability that the $(2k+2)$-nd part belongs to $\lambda^{(1)}$ tends to $1/2$. In particular, the probability that the largest part belongs to $\lambda^{(1)}$ tends to

$$c_0=\frac{1}{2}\left(1+\prod_{i=0}^{\infty}\frac{1-q^{2i+1}}{1-q^{2i+2}}\right).$$

The statement is presented conditionally on an identity for expected central characters that had been verified only through $k=6$; its general validity and the resulting asymptotics remain open.