Duncan–Steingrímsson's Bell-number conjecture for modified ascent sequences

A modified ascent sequence is a combinatorial sequence of integers equipped with the ascent-pattern avoidance notion used here; a pattern is avoided when it does not occur as a pattern in the sequence. The six patterns under consideration are

$$212,\;1212,\;2132,\;2213,\;2231,\;2321.$$

Duncan–Steingrímsson's conjecture. On modified ascent sequences, these six patterns are all Wilf-equivalent, and the enumeration of modified ascent sequences avoiding any one of them is given by the Bell numbers. Moreover, the distribution of the number of ascents is the reverse of the distribution of blocks on set partitions.

The conjecture concerns pattern avoidance in Fishburn-type combinatorial structures and predicts both a common Bell-number enumeration and a refined correspondence with set partitions. The paper describes the corresponding Fishburn permutations by pattern avoidance and presents the work as mostly devoted to proving this conjecture; the supplied text does not state whether every part has been resolved.