The Motzkin enumeration conjecture for good tableaux

Let $T$ be a standard tableau, and call $T$ good when every permutation $w$ satisfying $\operatorname{Q}(w)=T$ is BBS good, equivalently when $T$ is the recording tableau of some good permutation. Motzkin conjecture. The number of size-$n$ good tableaux is equal to the $n$th Motzkin number. Motzkin numbers also enumerate several classical objects, but the asserted enumeration of good recording tableaux remains open.