Relative rigidity conjecture for post-critically finite regular polynomial endomorphisms

Let $X/\mathbb{C}$ be an affine curve, let $\sigma:X\to\mathsf{RPE}^{N}_{d}$ land entirely in the PCF locus, and suppose that $\pi\circ\sigma:X\to\mathsf{M}^{N-1}_{d}$ is constant. Then $\sigma$ is constant.

Relative rigidity conjecture. Every such family of post-critically finite maps is constant.

This conjecture proposes relative rigidity of PCF maps in the fibres of the projection from regular polynomial endomorphisms to the moduli space of their restrictions to the invariant hyperplane. The paper proves the analogous statement for the fibre above the $d$th power map, but gives no resolution for general fibres.