Convex ribbon disks, quasipositivity, and Lagrangian sliceness

We introduce a version of convex surface theory for immersed ribbon surfaces with Legendrian boundary. As an application, we establish a special case of K3: Problem 1.70 on sharpness in the slice-Bennequin inequality implying quasipositivity, and a special case of a conjecture of Cornwell, Ng, and Sivek on characterizing Lagrangian sliceness. Specifically, we prove that if $K$ is a knot that bounds a "star-shaped" ribbon disk and satisfies $\overline{\mathrm{sl}}(K) = -1$ (resp.\ $\overline{\mathrm{tb}}(K) = -1$), then $K$ is quasipositive (resp.\ Lagrangian slice). In fact, we prove these statements for any fixed transverse (resp.\ Legendrian) representative.