  In 2011, Duncan and Steingr\'imsson conjectured that modified ascent
sequences avoiding any of the patterns 212, 1212, 2132, 2213, 2231 and 2321 are
counted by the Bell numbers. Furthermore, the distribution of the number of
ascents is the reverse of the distribution of blocks on set partitions. We
solve the conjecture for all the patterns except 2321. We describe the
corresponding sets of Fishburn permutations by pattern avoidance, and leave
some open questions for future work.
