Visibility-preserving convexifications using single-vertex moves

Devadoss asked: (1) can every polygon be convexified so that no internal visibility (between vertices) is lost in the process? Moreover, (2) does such a convexification exist, in which exactly one vertex is moved at a time (that is, using {\em single-vertex moves})? We prove the redundancy of the "single-vertex moves" condition: an affirmative answer to (1) implies an affirmative answer to (2). Since Aichholzer et al. recently proved (1), this settles (2).