Down-degree homomesy conjecture for trapezoid posets

Let $\mathscr T(a,b)$ be the trapezoid poset, let $J(\mathscr T(a,b))$ be its set of order ideals, and let $\operatorname{ddeg}$ denote the down-degree statistic. A probability distribution $\mu$ on $J(\mathscr T(a,b))$ is toggle on antichains-symmetric if, for every antichain $A$ of $\mathscr T(a,b)$, the expectation of the antichain-toggleability statistic satisfies

$$\mathbb E[\mu;\mathcal T_A]=0.$$

Down-degree homomesy conjecture. For every toggle on antichains-symmetric distribution $\mu$ on $J(\mathscr T(a,b))$,

$$\mathbb E[\mu;\operatorname{ddeg}]=\frac{ab}{a+b}.$$

This weakens the corresponding statement for rectangles because not every toggle-symmetric distribution on trapezoid-poset ideals has the same expected down-degree. In particular, it predicts the stated constant for the distributions arising uniformly from rowmotion orbits, but the source gives no resolution of the conjecture.