Genus-zero bridge trisection conjecture for properly embedded surfaces

Let $S$ be a properly embedded smooth surface in $B^4$. Genus-zero bridge trisection conjecture. Every properly embedded smooth surface $S$ in $B^4$ has a genus $0$ bridge trisection. Bridge trisections extend the study of trisections to knotted surfaces, and a genus-zero bridge trisection would provide a particularly simple description of every such surface. The paper proposes this definition for surfaces with boundary and notes that it should be possible to adapt known results to establish the conjecture, but does not prove it.