After some breeding, I found evidence that proves otherwise. The judge told me that my Trapinch had perfect ( "flawless") Defense as it's highest stat, however an unexpectedly large percentage of Trapinch's eggs possessed perfect Special Defense as their highest stat. This was odd, because neither parent had a perfect IV in Special Defense according to the IV judge. So, I checked both parents' stats and entered them in an IV calculator. It turns out that this Trapinch had perfect IVs in both Defense and Special Defense! The judge did not mention anything about the perfect Special Defense.
I concluded that the judge will report only one highest stat in the case where there is a tie between two or more stats. This makes breeding harder, seeing as how you cannot fully determine a pokemon's perfect stats until you hatched the egg, leveled up the pokemon and run its stats through an IV calculator.
I'm thinking the game copies Gen 3's and Gen 4's stat judges who "randomly choose which one to mention." Oddly, the EV judge seems to mention a random stat when you present to her an egg, or any pokemon with zero EV's in each stat.