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.