Conjecture on arbitrarily long zero gaps in the privileged complexity of the Thue–Morse word

Let $t$ be the Thue–Morse word, and let $\mathpzc{A}_n(t)$ denote the number of privileged factors of $t$ of length $n$. A zero gap is a consecutive interval of lengths on which $\mathpzc{A}_n(t)=0$. Zero-gap conjecture. There exist arbitrarily long, but not infinite, zero gaps in the values of $\mathpzc{A}_n(t)$. The conjecture is based on computer searches and observed gaps of zeros, including a gap for every $n$ with $189\leq n\leq257$; the existence of arbitrarily long finite gaps remains open.