The Kreweras interval conjecture for tree-like chord diagrams

Let $C_n(T_{\geqslant 3},B_{\geqslant 3})$ be the set of connected chord diagrams of size $n$ avoiding all top and bottom cycles of size at least $3$, and let $\textnormal{Int}(\mathcal{L}^{K}_{n})$ denote the intervals of the Kreweras lattice. The Kreweras interval conjecture. There is a bijection

$$C_n(T_{\geqslant 3},B_{\geqslant 3})\cong \textnormal{Int}(\mathcal{L}^{K}_{n}),$$

and both sets have cardinality $\frac{1}{2n+1}\binom{3n}{n}$. This conjecture proposes a Catalan-poset interpretation for these tree diagrams; it is based on manual computations for small $n$, and its general validity remains open.