Artinian Geller conjecture for subalgebras of principal ideal algebras

Let $K$ be an algebraically closed field of characteristic zero, and let $A$ and $B$ be $K$-algebras essentially of finite type over $K$. Assume that $B$ is a finite-dimensional principal ideal $K$-algebra, meaning that every ideal of $B$ is principal. Artinian Geller conjecture. If $A$ is a subalgebra of $B$ and the map

$$K_2(A)\longrightarrow K_2(B)$$

is injective, then $A$ is also a principal ideal algebra.

This is an Artinian analogue of Geller's conjecture, and the paper proves that it implies Geller's conjecture. Its general validity is not established in the supplied text.