Power-of-two block conjecture for Ulam words

Let $u$ be a binary word of length $n$ of the form $0^a1^{2^k}0^b$, where $a,b,k\in\mathbb{Z}_{\geq1}$. Power-of-two block conjecture. The word $u$ belongs to $\mathscr{U}$ if and only if

$$n\equiv1\pmod{2^k}.$$

The claim would describe membership in the Ulam-word set for a structured family of words with a central block of $2^k$ ones. The source says only that this may be demonstrable; no proof or resolution is provided.