Let $\Gamma_g$ be a surface group of genus $g \geq 2$, that is we have a presentation: $$\Gamma_g = \langle x_1,y_1 \dots, x_g,y_g \vert \prod_{i = 1}^g [x_i,y_i] = 1\rangle$$

Let $H \leq \Gamma_g$ be a finitely generated subgroup.

Must $H$ be closed in the profinite topology on $\Gamma_g$?

That is, must there be a family of finite index subgroups of $\Gamma_g$ intersecting in $H$?

If the answer is no, then must there be for any $M \in \mathbb{N}$ a subgroup $H \leq K \leq \Gamma_g$ with $\infty > [\Gamma_g : K] \geq M$?