Configuration-count conjecture for merges of vincular patterns

Let $\sigma$ be a vincular pattern with $j$ blocks, let $k$ denote its length, and let $M_{\underline{\sigma}}$ be the relevant maximum common-block size. For $1\leq l\leq M_{\underline{\sigma}}$, let $c_{\sigma,l}$ be the number of possible configurations for a merge of two copies of $\underline{\sigma}$ in which the minimum size of the two merged blocks is $l$. Let $b'_{\sigma}(2k-l,2j-1)$ be the number of merges of two copies of $\underline{\sigma}$ with $l$ common elements and $2j-1$ blocks that correspond to one of these $c_{\sigma,l}$ configurations.

Configuration-count conjecture. For every vincular pattern $\sigma$ with $j$ blocks and every $1\leq l\leq M_{\underline{\sigma}}$,

$$b'_{\sigma}(2k-l,2j-1)>\frac{\binom{2k-l}{k}}{k_{(l)}}c_{\sigma,l}.$$

The authors state that this conjecture would imply the relevant inequality for vincular patterns, and consequently the associated theorem and central limit theorem for vincular patterns. The conjecture is presented without a resolution in the supplied text.