The no-connected-hypersurfaces conjecture for Jacobi spectral curves

Let the parameter space of finite Jacobi pencils be coordinatized by $a_1,\ldots,a_n,b_1,\ldots,b_{n-1}$, and call the chain connected when $b_1\cdots b_{n-1}\ne0$. The reducible locus consists of parameters for which the spectral polynomial $\chi_n(\lambda,w)$ is reducible.

No-connected-hypersurfaces conjecture. For every $n\ge4$, every codimension-one component of the reducible locus is one of the cutting hyperplanes $b_i=0$. Equivalently, the reducible locus restricted to the connected stratum $b_1\cdots b_{n-1}\ne0$ has codimension at least two. Scalar-diagonal strata have codimension $m-1$ for blocks of length $m$ and therefore do not contradict this prediction for $m\ge3$.

The claim predicts that connected reducibility has no divisorial component. The source presents the codimension assertions as expected and does not state a proof or a resolution.