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We propose in this article a conjecture on the necessary and sufficient conditions of compatibility: Given a set $\\mathcal{C}$ of $r$-states full characters, there exists a function $f(r)$ such that $\\mathcal{C}$ is compatible iff every set of $f(r)$ characters of $\\mathcal{C}$ is compatible. Some previous work showed that $f(2)=2$, $f(3)=3$ and $f(r) \\ge r-1$. Gusfield et al. 09 conjectured that $f(r) = r$ for any $r \\ge 2$. 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