Subword-complexity difference conjecture for the generalized Thue–Morse words

For each integer $k\geq 1$, let ${\bf x}_k$ be the infinite word defined as the limit of the locally catenative sequence described above, and let $\rho_{{\bf x}_k}(n)$ denote its subword complexity. The subword-complexity difference conjecture for ${\bf x}_k$. For all sufficiently large $n$, the first difference

$$\rho_{{\bf x}_k}(n+1)-\rho_{{\bf x}_k}(n)$$

takes only the values $4k-2$ and $4k$. The cases $k=1$, $k=2$, and $k=3$ are supported by known results, but the assertion for arbitrary $k$ remains open.