Consider a race between two Poisson distributed random variables. Conjecture: The race is more likely to be tied when the one behind gets faster.
If $N_1$ and $N_2$ are independent Poisson-distributed random variables with means $\mu_1$ and $\mu_2$, then $K=N_1-N_2$ follows the Skellam distribution. The probability that $N_1$ and $N_2$ are equal ("the race ends in a tie") is $p(K=0;\mu_1,\mu_2)$.
$\textbf{Claim.}$ $\;$ If $\,$ $0<\mu_1' < \mu_1'' < \mu_2-1,$ then $\,$ $p(0;\mu_1',\mu_2) < p(0;\mu_1'',\mu_2)$.
But: Is this claim true?
Update: A friend and co-author has found a proof. I plan to update this post with the argument soon. Thank you to the commentators.