Veronese projection conjecture for the constructed Lefschetz bipencil

A Lefschetz bipencil is a pair of compatible braid and fiber monodromies whose associated fibration is obtained by projecting a projective variety to a projective plane. In this section, the Lefschetz bipencil is constructed from explicit monodromy data, and let $v_2\colon \mathbb{P}^3\hookrightarrow\mathbb{P}^9$ denote the degree-$2$ Veronese embedding and $\pi\colon\mathbb{P}^9\dashrightarrow\mathbb{P}^2$ a generic projection.

Veronese projection conjecture. The Lefschetz bipencil constructed in this section is isomorphic to the composition

$$\pi\circ v_2\colon \mathbb{P}^3\hookrightarrow\mathbb{P}^9\dashrightarrow\mathbb{P}^2.$$

This conjecture identifies the combinatorially constructed bipencil with the one arising from a generic projection of the degree-$2$ Veronese embedding. It arises from an adaptation of Moishezon–Teicher's methods, but several lemmas needed for the degeneration and regeneration procedure remain unproved.