The 3/4 conjecture for fix-free codes with at most three distinct codeword lengths

—The 3 / 4 conjecture was posed 25 years ago by Ahlswede, Balkenhol, and Khachatrian, and states that if a multiset of positive integers has Kraft sum at most 3 / 4 , then there exists a code that is both a preﬁx code and a sufﬁx code with these integers as codeword lengths. We prove that the 3 / 4 conjecture is true whenever the given multiset of positive integers contains at most three distinct values.