Classification of $(2,0;2)$-bridge trisections

A $(2,0;2)$-bridge trisection is a bridge trisection with parameters $(2,0;2)$. The examples in Example~ describe the planar surfaces in $S^3$ bounded by $(2,n)$-torus links, with the link represented as a 2-braid.

Classification conjecture. Every $(2,0;2)$-bridge trisection is diffeomorphic to one of those described in Example~ and Figure~.

The claim would classify all bridge trisections with these parameters by the displayed family associated to $(2,n)$-torus links. The source provides no resolution of this classification question.