I think the following is true, but haven't came up with a proof myself. Thanks in advance!

Let $G$ be a semisimple (to avoid more words) algebraic group over $\mathbb{C}$. Write $F=\mathbb{C}((t))$ and $\mathcal{O}=\mathbb{C}[[t]]$. Is it true that a maximal solvable $\mathcal{O}$-finite Lie subalgebra of $\mathfrak{g}(F)$ is an Iwahori subalgebra? In other words, is it true that all such Lie subalgebras are conjugate?

(feel free to replace $F$ by your favorite non-archimedean local field)

I think the following is true, but haven't came up with a proof myself. Thanks in advance!

Let $G$ be a semisimple (to avoid more words) algebraic group over $\mathbb{C}$. Write $F=\mathbb{C}((t))$ and $\mathcal{O}=\mathbb{C}[[t]]$. Is it true that a maximal solvable Lie subalgebra of $\mathfrak{g}(F)$ that is also a finitely generated $\mathcal{O}$-module must be an Iwahori subalgebra? In other words, is it true that all such Lie subalgebras are conjugate?

(feel free to replace $F$ by your favorite non-archimedean local field)