Is there a finitely presented group $G$ where every noncyclic subgroup $H$ of $G$ that is generated by $2$ elements is not finitely presented?
Context:
I was wondering about subgroups of finitely presented groups - I know that they need not be finitely generated (for example, the kernel of the abelianization $F_2 \to \mathbb{Z}^2$) and even when they are finitely generated, they need not be finitely presented (for example, the kernel of $F_2 \times F_2 \to \mathbb{Z}$ sending all the obvious generators to $1$ - see this question). I was wondering if there is always some finitely presented proper nontrivial subgroup and then I realized, of course cyclic subgroups need to be ruled out. So then I thought about the above question. Then I realized that I had no idea how to approach it - hence the question.