The divisibility conjecture for morphisms of n-circular bayonet codes

Let an $n$-cbc $X$ be a bayonet code of the type defined in the source, and let $d$ be an integer prime to $n$. For integers $d,j,n$, let $\overline{dj}^{n}$ denote the representative of $dj$ modulo $n$. Define

$$\varphi_d(X):=\left\{a^i b a^{\overline{dj}^{n}}\text{ such that }a^i b a^j\in X\right\}.$$

Divisibility conjecture for n-circular bayonet codes. For any $n$-cbc $X$ and any $d$ prime to $n$, the set $\varphi_d(X)$ is an $n$-cbc.

The source says that this conjecture was proposed in earlier work and places it in the study of divisibility conditions for an $n$-cbc containing a given bayonet code. No resolution is supplied in the provided text.