Could Quasar Lensing Time Delays Hint to a Core Component in Halos, Instead of $H_0$ Tension?

The time delay measured between the images of gravitationally lensed quasars probes a combination of the angular diameter distance to the source-lens system and the mass density profile of the lens. Observational campaigns to measure such systems have reported a determination of the Hubble parameter $H_0$ that shows significant tension with independent determination based on the cosmic microwave background (CMB) and large-scale structure (LSS). We show that lens mass models that exhibit a cored component, coexisting with a cusp, probe a degenerate direction in the lens model parameter space, being an approximate mass sheet transformation. This family of lens models has not been considered by the cosmographic analyses. Once added to the model, the cosmographic error budget should become dependent on stellar kinematics uncertainties. We propose that a core component coexisting with a cusp could bring the lensing measurements of $H_0$ into accordance with the CMB/LSS value.