Peczarski's Gold Partition Conjecture for finite posets

Let $P$ be a finite poset that is not a chain. Consider two consecutive comparisons, and let $t_0=e(P)$, where $e(P)$ is the number of linearizations of $P$; let $t_1$ be the number of linearizations of the poset after the first comparison, and let $t_2$ be the number of linearizations after both comparisons. Gold Partition Conjecture. There are two consecutive comparisons such that, regardless of their results,

$$t_0>t_1+t_2.$$

This conjecture implies the 1/3-2/3 conjecture. The source attributes it to Marcin Peczarski; the general conjecture remains open, despite being known through the implication and the many established cases of the 1/3-2/3 conjecture.