Shallit's factor-complexity conjecture for Thue-Morse-like sequences

Let $k\geq 1$, and let $\mathbf{x}_k$ be the Thue-Morse-like sequence obtained from the family described in the source. For a binary sequence, its factor complexity $p(n)$ is the number of distinct factors of length $n$; its first difference is $p(n+1)-p(n)$. Shallit's factor-complexity conjecture. For sufficiently large $n$, the first difference of the factor complexity of $\mathbf{x}_k$ takes only the values

$$4k-2\quad\text{and}\quad 4k.$$

This conjecture extends the known cases for the Thue-Morse sequence, the Fibonacci-Thue-Morse sequence, and the Allouche--Johnson sequence. The source gives no resolution, so the conjecture remains open.