The square-root conjecture for characteristic-polynomial coefficients of special D-partitions

Let $[p]$ be a special D-partition, and write the characteristic polynomial in the form

$$\operatorname{charpol}(\phi)=\lambda^{2N}+\sum_{k=1}^N t^{\chi_{2k}}\bigl(c_{2k}+O(t)\bigr)\lambda^{2(N-k)},$$

where, if $[p]$ has parts $p_i$, then $\chi_{2k}=j$ whenever $\sum_{r=1}^{j-1}p_r<2k\leq\sum_{r=1}^j p_r$. For every part $p_{2i}$ satisfying $p_{2i}>p_{2i+1}$, set $2k_i=\sum_{j=1}^{2i}p_j$.

Square-root conjecture. Then $c_{2k_i}=(a_{k_i})^2$, where $a_{k_i}$ is a polynomial on the Lie algebra.

The coefficients $c_{2k}$ are initially known to be polynomials on the Lie algebra of degree $\chi_{2k}$; the conjecture asserts a stronger square-factorization property for the coefficients indexed by the specified breaks in a special D-partition.