Following the evolution under a non-Hermitian Hamiltonian (nHH) involves significant probability loss. This makes various nHH effects impractical in the quantum realm. In contrast, Lindbladian evolution conserves probability, facilitating observation and application of exotic effects characteristic of open quantum systems. Here we are concerned with the effect of chiral state conversion: encircling an exceptional point, multiple system states are converted into a single system eigenmode. While for nHH the possible converted-into eigenmodes are pure states, for Lindbladians these are typically mixed states. We consider hybrid-Liouvillian evolution, which interpolates between a Lindbladian and a nHH and enables combining the best of the two worlds. We design adiabatic evolution protocols that give rise to chiral state conversion with pure final states, no probability loss, and high fidelity. Furthermore, extending beyond continuous adiabatic evolution, we design a protocol that facilitates conversion to pure states with fidelity 1 and, at the same time, no probability loss. Employing recently developed experimental techniques, our proposal can be implemented with superconducting qubit platforms.