Avoidability threshold for powers in odd-difference arithmetic progressions

Let an infinite word be a sequence over a 4-letter alphabet. An $r$-power is a word consisting of repetitions of a nonempty block with exponent $r$, and a word avoids $r$-powers in arithmetic progressions of odd difference if no such factor occurs along an arithmetic progression whose common difference is odd. The power-avoidability conjecture. For every real number $r<2$, $r$-powers are not avoidable in arithmetic progressions of odd difference over a 4-letter alphabet. The construction above gives uncountably many 4-letter words avoiding squares in arithmetic progressions of odd difference, but each such word contains $r$-powers for every $r<2$; a backtracking search confirms the conjecture for $r\leq 7/4$, while the general range $7/4<r<2$ remains open.