Unimodality conjecture for tableau types and the statistic $a_{(m2^a1^b)}$

Let $m\in\\{3,4\\}$, let $S$ be a standard tableau of size $m$, and let $s$ be a sequence of $a$ standard tableaux of size $2$ and $b$ standard tableaux of size $1$. For a standard tableau $T$ of size $m+2a+b$, write $type_{(m2^a1^b)}(T)=(S,s)$ and let $a_{(m2^a1^b)}(T)$ be its associated statistic. Define

A_{(S,s)}^i=\\#\\{T\mid T\in ST^{m+2a+b},\\ type_{(m2^a1^b)}(T)=(S,s),\\ a_{(m2^a1^b)}(T)=i\\}.

Unimodality conjecture. For each fixed $(m2^a1^b)$-type $(S,s)$, the sequence

$$A_{(S,s)}^*=(A_{(S,s)}^0,A_{(S,s)}^1,A_{(S,s)}^2,\ldots)$$

is unimodal; equivalently, the number of tableaux with fixed type and statistic value $i$ increases to a maximum and then decreases as $i$ ranges over all possible values. The claim applies for $m=3$ or $m=4$.

This is presented as an analogue of Conjecture 4.8 of the cited work and is supported in the source by experimental calculations. The source gives no proof or resolution, so the conjecture remains open.