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Multiple image gravitational lensing systems with measured time delaysprovide a promising one-step method for determining $H_0$. MACS J1149, whichlenses SN Refsdal into a quad S1-S4, and two other widely separated images, SXand SY, is a perfect candidate. If time delays are pinned down, the remaininguncertainty arises from the mass distribution in the lens. In MACS J1149, themass in the relevant lens plane region can be constrained by (i) many multipleimages, (ii) the mass of the galaxy splitting S1-S4 (which, we show, iscorrelated with $H_0$), (iii) magnification of SX (also correlated with $H_0$),and (iv) prior assumptions on the mass distribution. Our goal is not toestimate $H_0$, but to understand its error budget, i.e., estimateuncertainties associated with each of these constraints. Using multiple imagepositions alone, yields very large uncertainty, despite the fact that theposition of SX is recovered to within $\!\le\!0.036$" (rms $\!\le\!0.36$") byGRALE lens inversion. Fixing the mass of the galaxy that splits S1-S4 reduces$1\sigma$ uncertainties to $\sim 23\%$, while fixing the magnification of SXyields $1\sigma$ uncertainties of $32\%$. We conclude that smalleruncertainties, of order few percent, are a consequence of imposing priorassumptions on the shapes of the galaxy and cluster mass distributions, whichmay or may not apply in a highly non-equilibrium environment of a mergingcluster. We propose that if a measurement of $H_0$ is to be consideredreliable, it must be supported by a wide range of lens inversion methods.

The role of multiple images and model priors in measuring $H_0$ from supernova Refsdal in galaxy cluster MACS J1149.5+2223