On the $K$-extensions between Serre weights for unramified ${\mathrm{G}\mathrm{L}}_3$

Let $p\ge 5$ be a prime number. Let $L$ be a finite unramified extension of ${\mathbb{Q}}_p$ with ring of integers ${\mathcal{O}}_L$ and residue field ${\mathbb{F}}_q$. Given two Serre weights for ${\mathrm{G}\mathrm{L}}_3({\mathbb{F}}_q)$, we prove that in most cases the extensions between them for ${\mathrm{G}\mathrm{L}}_3({\mathcal{O}}_L)$ modulo the center coincide with their ${\mathrm{G}\mathrm{L}}_3({\mathbb{F}}_q)$-extensions.