The eigenvalue convex-hull conjecture for Toeplitz-Hessenberg matrices

Let $A$ be a nonnegative substochastic matrix, meaning

$$\sum_i A_{i,j}\leq 1\quad\text{and}\quad\sum_j A_{i,j}\leq 1$$

for all $i$ and $j$. Suppose that above the diagonal, $A$ has nonzero entries only at distance $1$ from the diagonal, while below the diagonal it has nonzero entries only at distances at most $k$, with the diagonal having distance zero. Eigenvalues of Toeplitz-Hessenberg Matrices conjecture. The eigenvalues of $A$ are contained in the convex hull of the $k+1$ roots of unity in the complex plane. The paper introduces this as a further open conjecture motivated by the trace conjectures and gives no resolution.