Secondary braid group conjecture for trigonal abelian-differential strata

Let $k$ be an integer for which the indicated trigonal stratum ${\mathbf P}{\mathcal H}^{tri}_{3k+1}(4k,2k)$ is defined, and let ${}^2Br\bigl(\sigma_1(\sigma_2\sigma_1)^{3k+2}\bigr)$ denote the secondary braid group associated with the displayed positive braid word. Secondary braid group conjecture. The orbifold fundamental group is isomorphic to

$$\pi_1^{orb}{\mathbf P}{\mathcal H}^{tri}_{3k+1}(4k,2k)\cong{}^2Br\bigl(\sigma_1(\sigma_2\sigma_1)^{3k+2}\bigr).$$

Moreover, this secondary braid group has the finite presentation

$$\left\langle t_1,\dots,t_{6k+3}\ \Bigg|\ \begin{array}{ll} t_i t_j=t_j t_i & \text{if } |i-j|>2 \\ t_i t_j t_i=t_j t_i t_j & \text{if } |i-j|\leq2 \\ t_i t_{i+1}t_{i+2}t_i=t_{i+1}t_{i+2}t_i t_{i+1} \end{array}\right\rangle.$$

The conjecture is motivated by interpolation between identifications of related orbifold fundamental groups with secondary braid groups and by the corresponding links at infinity; the source gives no proof or resolution.