Root-distribution conjecture for Kadar–Yu Chebyshev polynomials

Fix an integer $l$ and let $\lambda\vdash l+2$. For each $k\geq -1$, let $P^{(\lambda)}_{l+4+k}(\alpha)$ be the associated polynomial.

Root-distribution conjecture. For each $k\geq -1$, the polynomial $P^{(\lambda)}_{l+4+k}$ is square free. Furthermore, for sufficiently large $k$, it has a fixed number of roots outside $[-2,2]$, and

$$\bigcup_{k=0}^{\infty}\{\alpha\in[-2,2]\mid P^{(\lambda)}_{l+4+k}(\alpha)=0\}$$

is dense in $[-2,2]$.

The claim summarizes the root patterns observed for several partition families. It predicts both eventual stability of the number of exterior roots and density of the interior roots.