The smallest eigenvalue of a circulant B-spline matrix

Let $C_m^d$ be the circulant associated with the degree-$d$ B-spline collocation matrix, with Toeplitz symbol coefficients $t_0^d,\ldots,t_r^d$, and let $\lambda_\mu(C_m^d)$ denote its smallest eigenvalue. The smallest-eigenvalue conjecture. The circulant $C_m^d$ is always positive definite, and the index $\mu$ of its smallest eigenvalue is always the integer nearest to $m/2$; explicitly,

$$\lambda_{\mu}(C_m^d)=\begin{cases}\displaystyle\lambda_{(m\pm1)/2}(C_m^d)=t_0^d+2\sum_{j=1}^r(-1)^jt_j^d\cos\left(\frac{\pi j}{m}\right),&m\text{ odd},\\[4pt]\displaystyle\lambda_{m/2}(C_m^d)=t_0^d+2\sum_{j=1}^r(-1)^jt_j^d,&m\text{ even.}\end{cases}$$

The statement is presented as a conjecture based on extensive numerical testing; no resolution is given in the supplied text.