Minimal exponential measure model in the post-Newtonian limit

We examine the post-Newtonian limit of the minimal exponential measure (MEMe) model presented in [J. C. Feng, S. Carloni, Phys. Rev. D 101, 064002 (2020)] using an extension of the parameterized post-Newtonian (PPN) formalism which is also suitable for other type-I minimally modified Gravity theories. The new PPN expansion is then used to calculate the monopole term of the post-Newtonian gravitational potential and to perform an analysis of circular orbits within spherically symmetric matter distributions. The latter shows that the behavior does not differ significantly from that of general relativity for realistic values of the MEMe model parameter $q$. Instead the former shows that one can use precision measurements of Newton's constant $G$ to improve the constraint on $q$ by up to $10$ orders of magnitude.