One-survivor conjecture for multitype urns on cycles

Let $C_k$ be the cycle graph of length $k$, and consider the $s$-type urn model on $C_k$, with balls of $s$ colours and each urn monochromatic. At each step, a uniformly chosen ball adds a ball of its colour to each neighbouring urn, with one-to-one annihilation against balls of another colour when necessary. A colour survives if it is present at arbitrarily large times. Cycle urn conjecture. For every $s,k\geqslant3$ and every finite initial configuration, the $s$-type urn model on $C_k$ almost surely has only one surviving colour. The corresponding two-type urn result is proved, but this multitype extension is not known; it would imply the related multitype growth result in the relevant setting.