Conjectures on the roots and interlacing of Lambert W-function polynomials

Let $L_n(X)$ and $L_{n,k}(X)$ be the polynomial families defined earlier in the paper, and let the roots of a polynomial be ordered decreasingly when indexed by $j$. For $k\leq 0$, interpret $L_{n,k}(X)$ as $L_n^{(1-k)}(X)$. Root and interlacing conjectures. The roots of $L_n(X)$ are all nonnegative real and are at most $n+\log(n)+\gamma+O(1/n)$; more generally, its $j$th root is asymptotic to

$$\frac{n}{j}+\sum_{j\leq m\leq n}\frac{1}{m}.$$

The roots of $L_n(X)$ and $L_{n+1}(X)$ interlace. More generally, for $k\geq 1$, the roots of $L_{n,k}(X)$ are all nonnegative real, and their maximum has an asymptotic expansion

$$n+\log(n)+\gamma+k-1+\frac{C_0(k)}{n}+\frac{C_1(k)}{n^2}+\cdots,$$

where $C_m(k)$ is a polynomial of degree $m$ in $k$, and $C_0(k)$ is independent of $k$; the same assertion holds for $k\leq 0$ under the interpretation above. These conjectures concern the zero distribution and asymptotic behavior of the polynomial families arising in the branch expansions of the Lambert $W$ function. The paper presents them as additional properties requiring proof; the notation and precise prior definitions of $L_n(X)$ and $L_{n,k}(X)$ should be checked against the source.