Explicit characterization of $\widehat{\mathfrak{g}}$-dominant tableaux for $n\le 4$ via $1$-$0$-slack recording tableaux in the quantum Littlewood-Richardson rule and other bijections

Previously we have explicitly characterized by certain linear inequalities the $\mathfrak{k}$-highest weight tableaux in the quantum Littlewood-Richardson (LR) rule produced by $1$-$0$-slack recording tableaux. Using the composition of promotion operators to defining the Naito-Suzuki-Watanabe bijection between $\mathfrak{k}$-highest weight tableaux and $\widehat{\mathfrak{g}}$-dominant tableaux, we now explicitly characterize by certain linear inequalities the $\widehat{\mathfrak{g}}$-dominant tableaux for $n\le 4$ produced by $1$-$0$-slack recording tableaux in the quantum Littlewood-Richardson rule. Since recording tableaux in the quantum Littlewood-Richardson rule are in natural bijection with Littlewood-Richardson-Sundaram tableaux, we relate our results with other bijections for the Naito-Sagaki conjecture.