The bounded-excess conjecture for generic two-row -modular matrices

For a positive integer $\Delta$, let $\operatorname{g}(\Delta,2)$ denote the maximum number of columns of a generic $\Delta$-modular integer matrix with two rows.

Bounded-excess conjecture. There is a constant $c>0$ such that

$$\operatorname{g}(\Delta,2)\leq\Delta+c$$

for all $\Delta>0$.

The question is motivated by computational results through $\Delta=450$, which show $\operatorname{g}(\Delta,2)\leq\Delta+6$ in that range. The supplied text does not establish whether a uniform constant exists.