Box Office Mojo has updated its release schedule to include "Little Women" for Christmas Day 2019. This position suggests that Sony expects the film to be an awards player (duh), as well as a potential holiday box office hit. The 1994 version was also released around the holidays, with the film's snowy settings a natural fit for a December premiere.
Of course this date is subject to change, but it's sort of a no-brainer for this level of project. Shooting is set to begin in October.