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Two sufficient conditions are given for the existence of binary fix-free codes (i.e., both prefix-free and suffix-free). Let L be a finite multiset of positive integers whose Kraft sum is at most 3/4. It is shown that there exists a fix-free code whose codeword lengths are the elements of L if either of the following two conditions holds: i) The smallest integer in L is at least 2, and no integer in L, except possibly the largest one, occurs more than 2min(L)-2 times. ii) No integer in L, except possibly the largest one, occurs more than twice. The results move closer to the Ahlswede-Balkenhol-Khachatrian conjecture that Kraft sums of at most 3/4 suffice for the existence of fix-free codes.