Bass–Quillen–Suslin conjecture on completing unimodular vectors

Let $R$ be a local ring. Assume either that $R$ is a regular local ring or that $1/r!\in R$. Let $T=(T_1,\ldots,T_n)$ and let $v$ be a unimodular vector over $R[T]$ of length $r+1$, meaning that its entries generate the unit ideal. Bass–Quillen–Suslin conjecture. The vector $v$ can be completed to an invertible matrix over $R[T]$.

This is a matrix-completion formulation related to the Bass–Quillen conjecture for projective modules. The supplied text gives no resolution status, so the conjecture is recorded as open.