A finitely generated group $G$ is called limit group if, for any finite subset $S\subset G$, there exists a homomorphism $f:G\to F$ (where $F$ is a free group of finite rank) so that the restriction of $f$ to $S$ is injective.
Let $H$ be a proper retract of $G$, i.e., a subgroup of $H\lneq G$ for which there exists a homomorphism $r:G\to H$ such that $r(h)=h$, for all $h\in H$.
By Louder's theorem, there is a function $D(n)$ such that if $F_n \to L_1 \to \cdots \to L_k$ is a sequence of proper epimorphism of limit groups, where $F_n$ is a free group with $n$-generator, then $k\leq D(n)$. Let $G$ be a limit group. In particular, take $L_1$ equal to $G$ which is a finitely presented (as a limit group) then there exists an integer $k_G$ such that for every sequence with proper retrctions $G=L_1 \to G_2 \to \cdots \to G_k$, we have $k\leq k_G$. If $H$ is a proper retract of a limit group $G$, then is $k_H \lneq k_G$?