Linear-length anti-power conjecture for aperiodic morphic words

Let $W$ be a sufficiently well-behaved aperiodic morphic word. A $k$-anti-power is a concatenation of $k$ consecutive pairwise distinct words of equal length.

Linear-length anti-power conjecture. There is a constant $C=C(W)$ such that, for all positive integers $i$ and $k$, $W$ contains a $k$-anti-power with blocks of length at most $Ck$ beginning at its $i$th position.

This conjecture proposes a uniform linear bound on the block length needed to find anti-powers at every position of a sufficiently well-behaved aperiodic morphic word. The phrase “sufficiently well-behaved” is intentionally left vague in the source, so the precise class of words for which the claim should hold remains open.