Bousquet-Mélou–Xin conjecture on non-D-finiteness of noncrossing partition generating functions

Let $k$-noncrossing set partitions be set partitions whose arc diagrams contain no $k$-crossing, and let their ordinary generating function count them by size. Bousquet-Mélou–Xin conjecture. For every $k>3$, the generating function of $k$-noncrossing set partitions is not D-finite. The case $k=3$ is known to be D-finite, while non-D-finiteness for larger $k$ remains open.