Growth problem for strong boundedness constants of special linear groups over number fields

For $n\in\mathbb{N}$, define

$$F(n):=\sup\left\{\frac{\Delta_k({\rm SL}_{n+2}(\mathscr{O}_K))}{k}\mid K\text{ a number field}\right\}.$$

Here $\mathscr{O}_K$ denotes the ring of integers of the number field $K$, and $\Delta_k$ is the strong boundedness invariant considered in the paper. Growth problem. What is the growth of the function

$$F:\mathbb{N}\to[0,+\infty),\qquad n\mapsto F(n)?$$

This is posed as an open problem about the dependence on the rank of the strong boundedness constants for $\operatorname{SL}_{n+2}(\mathscr{O}_K)$, uniformly over number fields. The preceding discussion gives linear bounds in the rank for global function fields, while the corresponding growth for number fields is left unresolved.