Unequal-rate single-survivor conjecture for competing urns on cycles

Let $C_N$ be a cycle, let $K\ge3$, and consider the $K$-type competing urn scheme with a nonzero initial configuration. For each type $j$, let balls of type $j$ have independent Poisson clocks ringing at rate $\lambda_j$. Unequal-rate single-survivor conjecture. The process has almost surely a single surviving type. The equal-rate case is established in the source paper, whereas the unequal-rate extension is presented as an open problem because the paper's methods do not readily handle multiple rate parameters.