The product-category conjecture for C and non-crossing pair partitions

Let $C$ be the category whose morphism sets are the finite diagram sets $C(k,l)$, and let $NC_2$ denote the category of non-crossing pair partitions. Let $NC_2\times NC_2$ be their product category. Product-category conjecture. The category $C$ is isomorphic to

$$NC_2\times NC_2.$$

Together with the Catalan-square cardinality conjecture, this would explain the observed count of elements of $C(0,2k)$, since Catalan numbers count non-crossing pair partitions. The source presents this as a further conjecture and gives no evidence of resolution.