New conjecture on balanced-sequence critical exponents

Let a balanced sequence over an alphabet of size $d$ be an infinite sequence in which, for any two factors of equal length, the number of occurrences of each letter differs by at most $1$. The critical exponent of a sequence is the supremum of the exponents of its nonempty factors, and the minimal critical exponent is the infimum of these values over balanced sequences on the given alphabet.

New conjecture. For every alphabet size $d\geq 11$, the minimal critical exponent of a balanced sequence over a $d$-letter alphabet equals

$$\frac{d-1}{d-2}.$$

The paper proves the lower bound and constructs sequences attaining it for every even $d\geq 12$. The conjecture therefore remains open for alphabets of odd size.