= Solution
Let $K=\ker f_\rho$ and $C=Y_j/\operatorname{im}g_\rho$. Consider the map from the arrow term of the <extension complex of quiver representations> to $\operatorname{Hom}_k(K,C)$ that restricts its $\rho$-component to $K$ and then takes the quotient in $Y_j$. This map is surjective: a map $K\to C$ can be lifted to $Y_j$ and extended from $K$ to $X_i$.
Every coboundary is killed by this map, since for $x\in K$,
$$
(\delta h)_\rho(x)=g_\rho h_i(x)-h_jf_\rho(x)=g_\rho h_i(x)\in\operatorname{im}g_\rho.
$$
It therefore induces a surjection
$$
\operatorname{Ext}^1_Q(X,Y)\twoheadrightarrow\operatorname{Hom}_k(\ker f_\rho,\operatorname{coker}g_\rho).
$$
Both vector spaces in the final <Hom functor> are nonzero, so its dimension is positive. Hence \b[$\operatorname{Ext}^1_Q(X,Y)\ne0$]. This is the <kernel-cokernel obstruction to splitting a quiver extension> and remains valid for loops and repeated arrows elsewhere in the <quiver>.
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