I will introduce the notion of nearly Lefschetz fibrations, which can be viewed as the complements of neighborhoods of positive multisections in bordered Lefschetz fibrations. We show that these objects are supported by canonical Stein structures and provide a tool to understand Lagrangian submanifolds in Weinstein domains. In particular, we show that in 4-dimensional Weinstein domains, all exact Lagrangian disks with Legendrian boundaries are regular, in the sense of Eliashberg-Ganatra-Lazarev. This is joint work in progress with Agniva Roy and Joseph Breen.