  Within a 2+1 flavor Nambu\textendash Jona-Lasinio model, we calculate the
curvature coefficients and check them against available lattice QCD
estimations. With the observation that the flavor mixing due to the `t Hooft
determinant term significantly affects the $\kappa_{2}^{S}$, we explore the
effect of $\mu_{S}$ on the $T-\mu_{B}$ crossover lines. With the novel
determination of negative $\kappa_{2}^{B}$ at large $\mu_{S}$, we advocate the
importance of studying the same in lattice QCD.
