On elliptic surfaces which have no 1-handles

Gompf conjectured that the elliptic surface $E(n{)}_{p,q}$ has no handle decomposition without 1- and 3-handles. We prove that each of the elliptic surfaces $E(n{)}_{5,6}$, $E(n{)}_{6,7}$, $E(n{)}_{7,8}$ and $E(n{)}_{8,9}$ has a handle decomposition without 1-handles for $n\ge 4$, $n\ge 5$, $n\ge 9$ and $n\ge 24$, respectively.