Delta-only distance conjecture for optimal mixed-integer program solutions

Let $A$ be the constraint matrix, let $\Delta$ be the largest absolute value of any determinant of a square submatrix of $A$, and let $I,J\subseteq[n]$ index the variables required to be integer in the mixed-integer programs $(I\text{-MIP})$ and $(J\text{-MIP})$. Assume that $(J\text{-MIP})$ has an optimal solution. Delta-only distance conjecture. There exists a function

$$f:\mathbb{Z}_{\ge 1}\to\mathbb{R}$$

such that, for every optimal solution $w$ of $(I\text{-MIP})$, there exists an optimal solution $z$ of $(J\text{-MIP})$ satisfying

$$\lVert w-z\rVert_{\infty}\le f(\Delta).$$

The paper proves a bound depending on $\Delta$ and $|I\cup J|$, and conjectures that the dependence on the number of integer variables can be removed entirely. It is not known whether distances between optimal solutions can always be bounded solely in terms of $\Delta$.