Simplified presentation conjecture for tangent conic-line arrangements

Let $S$ be a conic-line arrangement with two tangent conics and $n$ lines which are tangent to these conics. Assume that the tangency points of $S$ are simple. Let $C_1$ be the conic with the maximal number $m$ of lines tangent to it. In the simplified presentation of $\boldsymbol{\pi_1(\mathbb{C}\mathbb{P}^2-S)}$, let $a$ denote the generator corresponding to $C_1$.

Simplified presentation conjecture. The presentation has the following properties:

1. It has $m$ relations of the form

$$(a x_i)^2=(x_i a)^2,$$

where the $x_i$ are some generators of the group. 2. For any two generators $y$ and $z$ different from $a$, one has

$$y a z a^{-1}=a z a^{-1}y.$$

In some computed presentations, relations of the second type may simplify to commutative relations between generators as a consequence of the other relations; the conjecture concerns the resulting simplified presentation and remains unverified in general.