Coefficientwise domination conjecture for even-rank partition-poset homology

Let $\alpha_{2n}$ be the Frobenius characteristic of the representation on maximal chains of $\Pi_{2n}$, and let $R_{2n}$ be the Frobenius characteristic of the homology of the even-rank subposet $\Pi_{2n}^e$. Write $a_i(2n)$ and $b_i(n)$ for the coefficients of $h_2^i h_1^{2n-2i}$ in $\alpha_{2n}$ and $R_{2n}$, respectively.

Coefficientwise domination conjecture. The symmetric function $\alpha_{2n}-R_{2n}$ is a nonnegative integer combination of homogeneous symmetric functions $h_\lambda$ indexed by partitions $\lambda$ of $n$ whose parts are $1$ or $2$. Equivalently,

$$b_i(n)\leq a_i(2n)\qquad\text{for all }2\leq i\leq n.$$

This is presented as a strengthening suggested by the available data; the source gives no resolution.