Vorst's conjecture on K-regularity and regularity

Let $R$ be a commutative ring of dimension $d$, essentially of finite type over a field $k$. The property $K_{d+1}$-regularity means that the natural map

$$K_{d+1}(R)\to K_{d+1}(R[x_1,\ldots,x_r])$$

is an isomorphism for every $r\geq 0$.

Vorst's conjecture. If $R$ is $K_{d+1}$-regular, then $R$ is regular.

Quillen proved that regular noetherian rings are $K_n$-regular for all integers $n$, and Vorst's conjecture predicts a converse in the stated finite-type setting. It was proven in dimensions at most one; the general case is not resolved in the supplied source.