Uniformity conjecture for elementary generation lengths over global fields

Let $K$ be a global field, let $S$ be a non-empty set of valuations of $K$ containing all its archimedean places, and let $R$ be the ring of all $S$-algebraic integers in $K$. Write $u_n(R)$ for the elementary generation length associated with $u_n$ for $obreak\operatorname{SL}_n(R)$.

Uniformity conjecture. There is a function $f:\mathbb{N}\to\mathbb{N}$ such that, for any $K$, $S$, and $R$ as above,

$$\nu_n(R)=f(n)$$

for all $n\geq 3$.

The conjecture would show that the precise elementary generation lengths depend only on $n$, not on the global field or the set of valuations. The preceding theorem gives a uniform quadratic upper bound, but the exact values of $\nu_n(R)$ remain unknown; in particular, the existence of an explicit universal formula or function is left open.