The lensing signal around galaxy clusters can, in principle, be used to test detailed predictions for their average mass profile from numerical simulations. However, the intrinsic shape of the profiles can be smeared out when a sample that spans a wide range of cluster masses is averaged in physical length units. This effect especially conceals rapid changes in gradient such as the steep drop associated with the splashback radius, a sharp edge corresponding to the outermost caustic in accreting halos. We optimize the extraction of such local features by scaling individual halo profiles to a number of spherical overdensity radii, and apply this method to 16 X-ray-selected high-mass clusters targeted in the Cluster Lensing And Supernova survey with Hubble. By forward-modeling the weak- and strong-lensing data presented by Umetsu et al., we show that, regardless of the scaling overdensity, the projected ensemble density profile is remarkably well described by an NFW or Einasto profile out to $R sim 2.5h^{-1}$Mpc, beyond which the profiles flatten. We constrain the NFW concentration to $c_{200c} = 3.66 pm 0.11$ at $M_{200c} simeq 1.0 times 10^{15}h^{-1}M_odot$, consistent with and improved from previous work that used conventionally stacked lensing profiles, and in excellent agreement with theoretical expectations. Assuming the profile form of Diemer & Kravtsov and generic priors calibrated from numerical simulations, we place a lower limit on the splashback radius of the cluster halos, if it exists, of $R_{sp}/r_{200m} > 0.89$ ($R_{sp} > 1.83h^{-1}$Mpc) at 68% confidence. The corresponding density feature is most pronounced when the cluster profiles are scaled by $r_{200m}$, and smeared out when scaled to higher overdensities.
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