Altman–Gaujal–Hordijk conjecture on densities of balanced sequences

Let $k\geq 3$, and let $\{r_1,\dots,r_k\}$ be a set of distinct positive reals with $r_1>\dots>r_k>0$. A sequence over a $k$-element set is balanced if, for every symbol, the numbers of occurrences in any two contiguous subsequences of equal length differ by at most $1$; its density is the limiting frequency of that symbol. Altman–Gaujal–Hordijk's conjecture. A balanced sequence with densities $r_1,\dots,r_k$ exists if and only if $\underline r=(r_1,\dots,r_k)$ is the $(k,2)$-power. The conjecture concerns the characterization of possible density sets for balanced sequences; its resolution status is not specified in the supplied text.