Binary-one-prefix Euler-transform recurrence conjecture for Lyndon words

A Lyndon word over $\{0,1\}$ is a word that is the unique minimum among all of its rotations. For $k\geq2$, let $\alpha=(1,0,0,\dots,0)$ be the prefix of length $k$, and let $\mathcal{L}_\alpha$ be the sequence counting Lyndon words of length $n$ with prefix $\alpha$. If $\pazocal{E}(\mathcal{L}_\alpha)=\{t_n^\alpha\}_{n=1}^\infty$ denotes its Euler transform, then binary-one-prefix recurrence conjecture.

$$t_{n+k}^\alpha=\sum_{i=0}^{k-1}t_{n+i}^\alpha$$

for all $n\geq1$. This specific recurrence is supported by the computational data reported in the source.