[math-fun] Question on cubefree sequences

This follows on a recent discussion of the lexically first cubefree sequence. I have computed what I believe to be a(0..10000) of the lexically first infinite cubefree word over alphabet {0,1} (treating 0 < 1). I started to submit this sequence, however, when I got to the comments section, I began to write Since an infinite cubefree word over {0,1} exists, a lexically first such word must exist, so this sequence is well-defined. Then I caught myself, because I realized the reasoning was unsound. For example, there is an infinite word over {0, 1} that includes the digit 1, but no lexically first such word. So, do we actually know that there is a lexically first cubefree word over {0, 1}?