Generalized Gompf conjecture for 4-dimensional 2-handlebodies

Let $W$ and $W'$ be $4$-dimensional $2$-handlebodies. Two such handlebodies are $2$-equivalent when they are related by a finite sequence of isotopies, handle slides, and creations or removals of cancelling handle pairs involving handles of index at most $2$. Generalized Gompf conjecture. Any two diffeomorphic $4$-dimensional $2$-handlebodies are $2$-equivalent. It is known that diffeomorphic handlebodies are $3$-equivalent, whereas whether they are always $2$-equivalent remains open.