Conjecture A on adjoining prescribed matrix blocks to division-ring embeddings

Let $n\in\mathbb{N}$ with $n>1$, and let $\varphi:F\to M_n(G)$ be a ring embedding, where $G$ and $F$ are division rings. Suppose that $a$ and $b$ are positive integers satisfying

$$a+b=n+1.$$

Let $A$ be an $a\times b$ matrix with entries in $G$. Conjecture A. There exist division-ring extensions $G\subseteq G^\prime$ and $F\subseteq F^\prime$, together with a ring embedding $\varphi^\prime:F^\prime\to M_n(G^\prime)$ such that $\varphi^\prime|_F=\varphi$ and $\operatorname{Im}(\varphi^\prime)$ contains a matrix whose upper-right $a\times b$ block is $A$. The conjecture would provide the extensions needed to construct tight embeddings for arbitrary parameters, and its validity would consequently disprove the pure semisimplicity conjecture; the paper establishes only partial cases, including those yielding tight embeddings for $n<5$.