Conjecture on modified ascent sequences and Bell-number enumerations

A modified ascent sequence is obtained from an ascent sequence by the modification process described in the source. For patterns $101$, $0101$, $1021$, $1102$, $1120$, and $1210$, consider modified ascent sequences avoiding one fixed pattern. The modified-ascent-sequence Bell conjecture. These six patterns are all Wilf equivalent on modified ascent sequences; the number avoiding any one of them is the $n$-th Bell number; and, more precisely, the number with $k$ ascents equals the number of set partitions of an $n$-element set with $n-k$ blocks. Thus these sequences are equinumerous with set partitions. The source presents this as a conjecture and gives no resolution.