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We prove two cases of a strong version of Dejean's conjecture involving extremal letter frequencies. The results are that there exist an infinite $\left({\frac{5}{4}}^{+}\right)$-free word over a 5 letter alphabet with letter frequency $\frac{1}{6}$ and an infinite $\left({\frac{6}{5}}^{+}\right)$-free word over a 6 letter alphabet with letter frequency $\frac{1}{5}$.