The antichain-sequence conjecture for naturally labeled posets

Let $(P,\omega)$ and $(Q,\tau)$ be naturally labeled posets, meaning that the labeling is compatible with the poset order. Write $(P,\omega)\sim(Q,\tau)$ when their $P$-partition generating functions are equal. For a poset $P$, let

$$\mathrm{anti}(P)=(a_1,\ldots,a_w),$$

where $a_i$ is the number of antichains in $P$ of size $i$ and $w$ is the width of $P$. Antichain-sequence conjecture. If $(P,\omega)\sim(Q,\tau)$, then

$$\mathrm{anti}(P)=\mathrm{anti}(Q).$$

This would strengthen the preceding result that equality of $P$-partition generating functions preserves the width of naturally labeled posets. The conjecture is presented as an extension of that corollary; the source gives no resolution.