Coloring probability inequality for minimal elements of posets

Let $P$ be a connected $n$-element poset with minimal elements $m_1,\ldots,m_s$. Define a coloring $c:P\to\mathcal{P}([s])$ by putting $i\in c(x)$ if $x\geq_P m_i$, where $[s]=\{1,\ldots,s\}$. Let $L_{n-2}$ denote the labeling obtained after $n-2$ applications of promotion, and let $\mathbb{P}$ denote the relevant probability distribution on labelings.

Coloring probability conjecture.

$$\mathbb{P}\left(c(L_{n-2}^{-1}(1))=\{s\}\,\middle|\,L(m_{s-1})=n\right)\geq\mathbb{P}\left(c(L_{n-2}^{-1}(1))=\{s\}\,\middle|\,L(m_s)=n\right).$$

The paper states that this inequality would imply the general bound for tangled labelings by showing that the number of tangled labelings increases when $m_{s-1}\lessdot m_s$; it remains open.