Not a problem at all. Linkedin is a networking site, not a resume or job application, so you’re welcome to publish anything you want. If you want to make it less misleading, you can add in the summary that your start date is X in case you feel like you’re being deceiving. Another option is to just put the year instead of the month, and you can do it now. There is no rule stating you have to put ALL your personal information out there.