The modulo-three conjecture for all-heads coin sequences

Let a coin sequence be a finite string of $0$'s and $1$'s, where $1$ denotes a heads-up coin, and say that a sequence is removable if all its coins can be removed under the linear arrangement with no gaps. For a positive integer $n$, write $1^n$ for the sequence consisting of $n$ heads-up coins.

Modulo-three conjecture. The sequence $1^n$ is removable if and only if

$$n\equiv 0\text{ or }1\pmod{3}.$$

The conjecture is motivated by the initial removable and non-removable examples, and is used to predict removability after reducing a general sequence to an all-heads sequence. The supplied text gives no resolution, so the conjecture remains open in this source.