This pharmaceutical-company handout (thanks, Jen!) about their Vitamin K2 includes a description of an experiment where subjects took the tablets on an empty stomach or with a meal. The variation in stomach contents made a huge difference: absorption, measured by blood concentration, was 7 times higher when the K2 was taken with a meal.
This is going to change how I take supplements, such as selenium, for the rest of my life. I’ve often taken them on empty stomach. I never realized the effect of doing so could be so large.