Composition conjecture for lasso contractions

Let $(\operatorname{\mathcal{L}},\eta)$ be a lasso on a category $\mathsf{C}$. For monomorphisms $f_1\colon X_1\hookrightarrow Y$ and $f_2\colon X_2\to Y_{/f}$ arranged as in the source diagrams, let $Y_{/f_1}$ and $(Y_{/f_1})_{/f_2}$ denote the successive contractions, with induced maps $\eta_{X_1}^{\#}$ and $\eta_{X_2}^{\#}$. Composition conjecture for lasso contractions. There is a subobject $f\colon X\hookrightarrow Y$ whose lasso contraction satisfies

$$Y_{/f}=(Y_{/f_1})_{/f_2}$$

and

$$\eta_X^{\#}=\eta_{X_2}^{\#}\eta_{X_1}^{\#}.$$

The claim would imply that lasso-contractions are closed under composition, a prerequisite for making the relevant category of objects and lasso-contractions functorial. The source explicitly says that this general composition question is unresolved, although it is straightforward for graph contractions.