Summary
From the following discussion, we conclude that: (a) the homogeneity of space implies (in special relativity) the homogeneity of time, and vice versa; (b) the assumption of homogeneity of space (or time) implies that the transformation formulae must be linear (see Equations (10) and (17)).
This last conclusion is contrary to Hoyer's affirmation in the paper quoted below.
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Ferrari, J.A. On the homogeneity of space and time in special relativity. Zeitschrift für Allgemeine Wissenschaftstheorie 22, 169–171 (1991). https://doi.org/10.1007/BF01801256
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DOI: https://doi.org/10.1007/BF01801256