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 \( \hat{\mathfrak g} \)-dominant tableaux, we now explicitly characterize by certain linear inequalities the \( \hat{\mathfrak g} \)-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.