Compactification of ALG and ALG* gravitational instantons

An $ALG$ or $ALG^*$ gravitational instanton may, after a suitable hyperKähler rotation, admit a compactification obtained by adding a canonical cycle to a rational surface. Compactification question for ALG and ALG gravitational instantons.* Do $ALG$ or $ALG^*$ gravitational instantons, up to a suitable hyperKähler rotation, admit compactifications obtained by adding a canonical cycle to rational surfaces other than the rational elliptic surface? This asks whether the known compactifications to rational elliptic surfaces exhaust the possible rational-surface compactifications. It is presented as a question arising from the study of compactifications, and the source gives no resolution.