Wolffhardt's projective-dimension conjecture for symmetric algebras

Let $R$ be a regular local ring and let $M$ be a finitely generated $R$-module. Write $\operatorname{Sym}_R(M)$ for its symmetric algebra, $\operatorname{UFD}$ for unique factorization domain, and $\operatorname{p.dim}(M)$ for the projective dimension of $M$.

Wolffhardt's conjecture. If $\operatorname{Sym}_R(M)$ is a $\operatorname{UFD}$, then

$$\operatorname{p.dim}(M)\leq 1.$$

The conjecture asks whether factoriality of the symmetric algebra over a regular local ring forces the module to have projective dimension at most one. The supplied text cites it as Conjecture 6.1.4 in Wolffhardt but gives no resolution.