Macdonald-polynomial odd-power logarithm conjecture

Let $P_{\lambda}(x;q,t)$ denote the Macdonald polynomial corresponding to a partition $\lambda$, and let

$$w(x;q,t)=\sum_{\lambda}\omega(\lambda)P_{\lambda}(x;q,t),$$

where the sum runs over all partitions $\lambda$. Write $p_k$ for the power-sum symmetric functions. Macdonald-polynomial odd-power logarithm conjecture. The expression

$$\log w(x;q,-1)+\sum_{n\geq1\text{ odd}}\frac1{2n}a^nc^np_{2n}+\sum_{n\geq2\text{ even}}\frac1{2n}a^{\frac{n}2}c^{\frac{n}2}\left(a^{\frac{n}2}c^{\frac{n}2}-2b^{\frac{n}2}d^{\frac{n}2}\right)p_{2n}$$

would belong to $\mathbb{Q}[[p_1,p_3,p_5,\dots]]$. The source presents this as a possible replacement of the Hall–Littlewood functions by Macdonald polynomials; it gives no evidence that the assertion has been proved or disproved.