Eigenvalue formula conjecture for completely factorizable operators

Let $T$ be a completely factorizable operator of degree $d$, meaning that

$$T[(x-z)^d]=\prod_{j=1}^d\bigl(x-(a_jz+b_j)\bigr).$$

Then the eigenvalue formula conjecture states that the first few eigenvalues $\lambda_0,\dotsc,\lambda_d$ are

$$\lambda_j=\frac{e_{d-j}(a_1,\dotsc,a_d)}{\binom{d}{j}},$$

where $e_k(y_1,\dotsc,y_d)$ is the $k$th elementary symmetric function in $d$ variables. This formula relates the spectral data of a completely factorizable operator to the elementary symmetric functions of the affine root parameters; the source gives the formula but no resolution or further contextual status.