The free-algebra pullback conjecture for elementary embeddings

Let $k_1,\ldots,k_n$ be elementary embeddings in $\mathcal{E}_{\lambda}$ such that $\mathcal{A}_{k_1,\ldots,k_n}$ is free, and let $\ell,p\in\mathcal{A}_{k_1,\ldots,k_n}$. The pullback $\ell^{-1}p$ is the embedding whose composition with $\ell$ is $p$, when such an embedding exists. Free-algebra pullback conjecture. If $\ell^{-1}p$ exists, then

$$\ell^{-1}p\in\mathcal{A}_{k_1,\ldots,k_n}.$$

The conjecture generalizes the $n=1$ case proved for the algebras considered in Theorem above; it asserts that pullbacks between elements of a free algebra remain inside that algebra whenever they exist. Its status is not established in the supplied text.