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We examine critically how tightly the density dependence of nuclear symmetry energy Esym(ρ) is con-strained by the universal equation of state of the unitary Fermi gas EUG(ρ) considering currently known uncertain-ties of higher order parameters describing the density dependence of the equation of state of isospin asymmetric nuclear matter. We found that EUG (ρ) does provide a useful lower boundary for the Esym(ρ) . However, it does not tightly constrain the correlation between the magnitude Esym(ρ0) and slope L unless the curvature Ksym of the symmetry energy at saturation densityρ0 is more precisely known. The large uncertainty in the skewness parameters affects the Esym(ρ0) versus L correlation by the same almost as significantly as the uncertainty in Ksym.