Common-eigenvalue conjecture for periodic tridiagonal matrices

Let $N$ be a positive integer and let $T^-$ and $T^+$ be the $(N-1)\times(N-1)$ tridiagonal matrices with

$$T^-_{jj}=2a\cos(2j\pi/N),\qquad T^+_{jj}=2a\cos[2(j-1)\pi/N]$$

for $j=1,\ldots,N-1$, and $T^\pm_{j,j+1}=T^\pm_{j+1,j}=1$. Assume $a>0$. Common-eigenvalue conjecture. The matrices $T^-$ and $T^+$ have a common eigenvalue if and only if $N$ is divisible by $4$; in that case, their common eigenvalue is $0$. This conjecture characterizes precisely when the associated discriminant equation has a double root, and would establish the asserted spectral pattern for these periodic recurrence coefficients.