  In 2023, Defant and Li defined the Ungarian Markov chain associated to a
lattice $L$. This Markov chain has state space $L$, and from any state $x \in
L$ transitions to the meet of $\{x\} \cup T$, where $T$ is a randomly selected
subset of the elements of $L$ covered by $x$. They defined $\mathcal{E}(L)$ to
be the expected number of steps until the maximal element $\hat{1}$ of $L$
transitions into the minimal element $\hat{0}$ in the Ungarian Markov chain,
and investigated its asymptotics when $L$ is the weak order on $S_n$, or the
$n$th Tamari lattice $\textrm{Tam}_n$. This paper resolves a conjecture of
Defant and Li and makes partial progress towards another, by proving that
$\mathcal{E}(S_n)$ is linear in $n$, and proving an $n^{1 - o(1)}$ lower bound
on $\mathcal{E}(\textrm{Tam}_n)$.
