Converse to the finite-global-dimension criterion for the singularity functor

With notation as above, let $R$ be a commutative ring, let $A$ be the algebra occurring in the construction of the singularity functor, and let $M$ be the corresponding maximal Cohen–Macaulay module. Assume that the singularity category $D_{\mathrm{sg}}(R)$ is idempotent complete. Converse finite-global-dimension conjecture. The module $M$ generates the singularity category $D_{\mathrm{sg}}(R)$ if and only if $A$ has finite global dimension. The question concerns whether surjectivity of the singularity functor forces finite global dimension of $A$; the source explicitly says that this converse is unclear to the author.