The stationary cover time of reversible chains on three and four states

Aldous and Fill asked whether every reversible Markov chain on $n$ states, started from its stationary law, has expected cover time at least $(n-1)H_{n-1}$, the value for the simple random walk on the complete graph $K_n$. We prove this for $n=3$ and $n=4$, for all reversible chains including those with holding, and show that equality holds only for the simple random walk on $K_n$. The case $n=3$ reduces to Schur's inequality. The case $n=4$ rests on an explicit polynomial identity of degree $16$ whose numerator becomes coefficientwise nonnegative after an ordered substitution on each of $30$ cones of edge orderings. Holding is handled for every $n$ by an occupation identity: if a chain without holding satisfies the bound and $E_iC\ge (n-1)H_{n-1}/2$ for every state $i$, then the bound survives any added holding. This pointwise condition holds for $n\le 4$.