Classification of totally irreducible $(2,0;1)$-bridge trisections

A $(2,0;1)$-bridge trisection is a bridge trisection of a surface with parameters $(2,0;1)$, and it is totally irreducible if it admits none of the relevant reducing moves. Two such trisections are considered equivalent up to diffeomorphism and mirroring.

Classification conjecture. There are exactly three, up to mirroring, totally irreducible $(2,0;1)$-bridge trisections.

These examples arise by puncturing the three unique $(3,1)$-bridge trisections of closed surfaces in $S^4$. The conjecture asks whether this list exhausts all totally irreducible trisections of the specified type.