We know Retro started developing a new game at the end of 2013. This was confirmed by them. On LinkedIn one of the employees put they were working on an "exploratory project for several years". After that they "moved on to another project and it was a secret"
That first project we can assume this employee moved on from the exploratory project around 2016.
We had Kotaku, I believe, say Retro was working on a project that "might be" potentially canned. Key words "might be", but they don't know and they said Retro is making Star Fox Grand Prix. If it lines up with the employee who could've been on the project Kotaku is talking about tells me they started Star Fox in 2016. I'm thinking middle of the year, but who knows.
Star Fox is supposed come out in 2019 according to the insiders and that would give it 3 years of development. That's plenty of time to develop a Star Fox racing game with an adventure mode.