An elementary generators-and-relations proof of the isomorphism defined by $\vartheta$

The paper considers the map $\vartheta$ arising in its comparison between the relevant generator-and-relation description and algebraic $K$-theory. Elementary proof conjecture. There should be an elementary proof that $\vartheta$ is an isomorphism, solely based on the generators and relations, and without truly touching algebraic $K$-theory. Such a proof would give a direct verification of the comparison result without using the machinery of algebraic $K$-theory.