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We study possible systematic effects on the values of the cosmologicalparameters measured through strong lensing analyses of the HFF galaxy clusterMACS J1149.5+2223. We use the observed positions of a large set ofspectroscopically selected multiple images, including those of supernova"Refsdal" with their estimated time delays. Starting from our reference modelin a flat $\Lambda$CDM cosmology, published in Grillo et al. (2018), we confirmthe relevance of the longest measurable time delay, between SX and S1, and anapproximately linear relation between its value and that of $H_{0}$. We performtrue blind tests by considering a range of time delays around its originalestimate of $345 \pm 10$ days, as an accurate measurement of this time delaywas not known at the time of analysis and writing. We investigate separatelythe impact of a constant sheet of mass at the cluster redshift, of a power-lawprofile for the mass density of the cluster main halo and of some scatter inthe cluster member scaling relations. Remarkably, we find that these systematiceffects do not introduce a significant bias on the inferred values of $H_{0}$and $\Omega_{\rm m}$ and that the statistical uncertainties dominate the totalerror budget: a 3% uncertainty on the time delay of image SX translates intoapproximately 6% and 40% (including both statistical and systematic $1\sigma$)uncertainties for $H_{0}$ and $\Omega_{\rm m}$, respectively. Furthermore, ourmodel accurately reproduces the extended surface brightness distribution of thesupernova host, covering more than $3 \times 10^{4}$ $HST$ pixels. We alsopresent the interesting possibility of measuring the value of theequation-of-state parameter $w$ of the dark energy density, currently with a30% uncertainty. We conclude that time-delay cluster lenses have the potentialto become soon an alternative and competitive cosmological probe.

On the accuracy of time-delay cosmography in the Frontier Fields Cluster MACS J1149.5+2223 with supernova Refsdal