It is a long-standing consensus in the literature that hard sphere perturbation theory (HSPT) is not accurate for dense soft sphere model liquids, interacting with repulsive r(-n) pair potentials for small n. In this paper, we show that if the intrinsic error of HSPT for soft sphere model liquids is accounted for, then this is not completely true. We present results for n = 4, 6, 9, 12 which indicate that, even first order variational HSPT can provide free energy upper bounds to within a few percent at densities near freezing when corrected for the intrinsic error of the HSPT.