Weinstein-trisection characterization of symplectic surfaces

Let $(X,\omega)$ be a symplectic $4$-manifold with a Weinstein trisection, and let $\mathcal{K}\subset (X,\omega)$ be a connected, oriented surface satisfying

$$\chi(\mathcal{K})=\langle c_1(\omega),[\mathcal{K}]\rangle-[\mathcal{K}]^2.$$

Weinstein-trisection characterization conjecture. The surface $\mathcal{K}$ is isotopic to a symplectic surface if and only if it is isotopic into transverse bridge position with respect to some Weinstein trisection of $(X,\omega)$.

The forward implication is suggested by the surrounding discussion, while the reverse implication follows from an adaptation of an earlier proposition in the paper. The statement is presented as a conjectural converse in the general Weinstein-trisection setting.