Prefix-closure conjecture for square-digit solutions

Let $\langle a_1',\dots,a_l'\rangle$ be a solution of length $l\ge 2$, where a solution is a digit sequence satisfying the paper's square-digit condition. Prefix-closure conjecture. The truncated sequence $\langle a_1',\dots,a_{l-1}'\rangle$ is also a solution. This asserts that every solution produces a solution after deleting its final digit; the source gives no resolution.