Quasivarieties of Sugihara semilattices with involution

We show that in the lattice of quasivarieties contained in the quasivariety generated by an $n$-element relatively subdirectly indecomposable Sugihara semilattice with involutions there exists a sublattice isomorphic to the lattice of ideals of a free lattice if and only if $n\geqslant3$. We also give two consequences of this result.