Coincidence point spectrum equality for disconnected digital images

Let $X$ and $Y$ be digital images. Assume that $X$ is disconnected and that $Y$ is totally disconnected with #Y>1.

Coincidence spectrum conjecture. The coincidence point spectra satisfy

$$CS_2(X,Y)=CS_3(X,Y)=\cdots=CS_i(X,Y)$$

for all $i\geq 2$.

The preceding results establish this equality when $X$ is connected, when $Y$ has one point, or when $Y$ contains an adjacent pair. The remaining case of disconnected $X$ and totally disconnected $Y$ with more than one point was stated as unresolved because the authors had not proved the equality.