Not a chance
Worked for them for twelve years, and I can categorically say "No". It's not the culture. This company runs like no other, store "Captains" (I was one) have pretty good autonomy to run things as they see fit and we had little meddling from Corporate. No need for a plane when the President can just call you on the phone - which he did on occasion. Buyers airline it - coach, but mostly let the mountain come to Mohamed.
The Albrecht family in Germany (the owners) might have a A/C but they are actually pretty low key and would likely use German Pilots. They only came over once a year anyway.
Trust me if there was a opening, I'd have been on it long ago. Good company to work for. Let that for this...