Large sum-free sets in ternary spaces

Conjecturally, if A⊆F3r is maximal sum-free and aperiodic (not a union of cosets of a non-zero subgroup), then |A|⩽(3r-1+1)/2. If true, this is best possible and allows one for any fixed ε>0 to establish the structure of all sum-free subsets A⊆F3r such that |A|>(1/6+ε)·3r.