The two-rectangle conjecture for Jack characters

Let $\lambda=(p_1,p_2)\times(q_1,q_2)$ be a multirectangular Young diagram consisting of at most two rectangles, and let $\operatorname{Ch}_\pi^{(\alpha)}(\lambda)$ and the orientability generating series be as in the measure-of-non-orientability conjecture. The two-rectangle conjecture. The measure-of-non-orientability conjecture is true for every such multirectangular Young diagram $\lambda$. The assertion would yield explicit formulas for the quadratic terms of Kerov polynomials for Jack characters; the source reports computer evidence but does not establish it.