Projectivity conjecture for cyclic local modules over endomorphism rings

Let ${\bf R}$ be a Noetherian local ring, let $E$ be a local ${\bf R}$-module, and let ${\Lambda}=\operatorname{End}_{\bf R}(E)$. Suppose that $E$ is ${\Lambda}$-cyclic and has finite projective dimension over ${\Lambda}$. Projectivity conjecture. Then $E$ is ${\Lambda}$-projective. The claim concerns when cyclicity and finite projective dimension force projectivity for local modules over their endomorphism rings; the supplied text does not state whether it is known or open.