The degree-eight distinct-diagonal reducibility conjecture for Jacobi spectral curves

Consider a connected degree-eight Jacobi pencil with pairwise distinct diagonal entries:

$$a_i\ne a_j\quad(i\ne j),\qquad b_1\cdots b_7\ne0.$$

A constant branch is a factor arising from a root $\lambda=-a_j$ that persists for all values of the spectral parameter, and reversal symmetry is the decomposition of a palindromic block into its symmetric and anti-symmetric parts.

Degree-eight distinct-diagonal test case. Every reducible connected degree-eight spectral curve is obtained from a constant branch or from reversal symmetry. In particular, there is no primitive non-palindromic $4+4$ or $3+5$ splitting with pairwise distinct diagonal entries.

The claim concerns the remaining Hensel-lifting possibilities after constant branches and reversal-symmetric factorizations are accounted for. The source describes these cases as expected and gives no proof or resolution.