I started dabbling in the Treasury auction market a year and a half ago. Since then, for Treasury notes, my experience has been that the interest rate frequently dropped a quarter point or so between the purchase deadline and the actual auctions, which closed a day or two later, around noon Eastern time.
A few times the rate stayed the same, once, it dropped almost half a point, but I do not recall the rate increasing into closing, which would have benefited me, the buyer (ie, the Greater Fool). Someone, somewhere, got a slice of the difference.