= Solution
The second <continuous-time Markov chain> runs in the reverse direction at rate $\rho$, as in the right diagram. Its nonzero <eigenvalues> have real part $-3\rho/2$, and its <transition probabilities> tend to $1/3$ for each vertex. <Independence> therefore gives
$$
p(t)=\sum_{v=A,B,C}P(X_t=v)P(Y_t=v)\longrightarrow3\left(\frac13\right)^2.
$$
Thus \b[the limit exists for every pair of starting vertices and equals $\boxed{1/3}$].
One can also see the limit directly without spectral multiplication. For fixed starting vertices, $X_t-Y_t$ modulo three advances with each jump of either flea. Its jump count is the sum of independent <Poisson processes> with rates $1$ and $\rho$, hence is Poisson with rate $1+\rho$ (multiply their generating functions). The formula from part (a), with $t$ replaced by $(1+\rho)t$ and the appropriate initial residue, gives $p(t)$ and the same limit.
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