The Catalan-square conjecture for connected chordal diagrams

Let $\mathcal{C}_{n}(T_{\geqslant 4},B_{\geqslant 4})$ be the set of connected chord diagrams of size $n$ avoiding top and bottom cycles of size at least $4$, and let $C_n$ denote the $n$th Catalan number. The Catalan-square conjecture. The cardinality of this set is

$$\left|\mathcal{C}_{n}(T_{\geqslant 4},B_{\geqslant 4})\right|=C_n^2.$$

The claim is one of several experimentally suggested enumerations for forbidden-subdiagram classes; it is stated as open in the source.