  Given a finite poset $\mathcal P$ and two distinct elements $x$ and $y$, we
let $\operatorname{pr}_{\mathcal P}(x \prec y)$ denote the fraction of linear
extensions of $\mathcal P$ in which $x$ precedes $y$. The balance constant
$\delta(\mathcal P)$ of $\mathcal P$ is then defined by \[ \delta(\mathcal P) =
\max_{x \neq y \in \mathcal P} \min \left\{ \operatorname{pr}_{\mathcal P}(x
\prec y), \operatorname{pr}_{\mathcal P}(y \prec x) \right\}. \] The
$1/3$-$2/3$ conjecture asserts that $\delta(\mathcal P) \ge \frac13$ whenever
$\mathcal P$ is not a chain, but except from certain trivial examples it is not
known when equality occurs, or even if balance constants can approach $1/3$.
  In this paper we make some progress on the conjecture by exhibiting a
sequence of posets with balance constants approaching
$\frac{1}{32}(93-\sqrt{6697}) \approx 0.3488999$, answering a question of
Brightwell. These provide smaller balance constants than any other known
nontrivial family.
