Conjectured equality of three-property and four-property maxima

Prove or refute that, for each n corresponding to the entries of Table \ref{tab:3prop}, the maximum number of 3-card sets achievable with n cards in the four-property SET deck equals the maximum number achievable with n cards in the three-property SET deck (i.e., the values reported in Table \ref{tab:3prop} for three properties also give the maxima for four properties).

Background

The authors compute, via exhaustive search, the exact maximum number of sets for the three-property SET deck for all n up to 27, yielding a table of values. Each such value is a lower bound for the four-property case by fixing one property.

They conjecture that these lower bounds are in fact tight for the four-property deck, at least for the n covered by the table. Establishing this equivalence would immediately provide exact maxima for the four-property deck up to n = 27; determining whether this extends beyond n = 27 would require additional data for the three-property case.