The low-crossing tri-plane classification conjecture for 2-knots

Let a tri-plane diagram be a diagram describing a bridge trisection of an embedded $S^2$ in $S^4$, and count its total crossings across the three constituent plane diagrams. Low-crossing tri-plane classification conjecture. If a tri-plane diagram for an embedded $S^2$ in $S^4$ has six or fewer total crossings, then it represents a bridge trisection of an unknotted sphere, the spun trefoil, or the $2$-twist spun trefoil. This is a classification conjecture for all tri-plane diagrams through six crossings; the supplied text does not state whether it has been proved or disproved.