Compact generation of local groups from compact generation of the universal group

Let $(\Delta,\mathcal{U}(\boldsymbol F))$ be a right-angled building with universal group determined by local groups $F_i$, indexed by $i\in I$. Assume that each local group $F_i$ is closed, that $\mathcal{U}(\boldsymbol F)$ is locally compact, and that $\mathcal{U}(\boldsymbol F)$ is compactly generated. Compact-generation converse. Does it then follow that each $F_i$ is compactly generated?

The preceding results establish one direction of the relationship between compact generation of the local groups and of the universal group, while the converse remains unresolved in the general setting.