The characteristic-polynomial and eigenvalue conjecture for unit-primitive matrices

Let $A_n$ be the $n\times n$ unit-primitive matrix, let $I_n$ be the $n\times n$ identity matrix, and let $P_n(x)$ be the polynomial defined by

$$P_n(x)=\sum_{k=0}^{n}P(n,k)x^k.$$

Let $U_n(x)$ denote the Chebyshev polynomial of the second kind, defined by $U_0(x)=1$, $U_1(x)=2x$, and $U_n(x)=2xU_{n-1}(x)-U_{n-2}(x)$. For positive integers $n$, define $\delta_{n,j}=2\cos\left(\frac{2j-1}{2n+1}\pi\right)$ for $j=1,\ldots,n$.

Unit-primitive matrix conjecture. The characteristic polynomial of $A_n$ is

$$f_{A_n}(x)=\det(xI_n-A_n)=x^nP_n(1/x),$$

and its $n$ eigenvalues are

$$w_{n,j}=U_{n-1}(\delta_{n,j}),\qquad j=1,\ldots,n.$$

These conjectural identities connect the characteristic polynomial and spectrum of the unit-primitive matrices with Chebyshev polynomials. The source later indicates that results concerning the eigenvalues and characteristic polynomial are proved, but the supplied parser status is unknown; the resolution should therefore be checked against the paper.