Ricci-flatness of Kähler-Ricci shrinkers on Fano cones

Let $X$ be a Fano cone, namely a polarized affine cone equipped with its cone structure and Reeb vector field. A Ricci-flat Kähler cone metric is, by definition in the source, a Kähler-Ricci shrinker when the fibration is the identity.

Fano-cone Ricci-flatness conjecture. On a Fano cone $X$, every Kähler-Ricci shrinker is a Ricci-flat Kähler cone metric.

This is a purely differential-geometric converse to the defining implication from Ricci-flat Kähler cone metrics to Kähler-Ricci shrinkers. The source gives heuristic arguments using degeneration to a tangent cone and monotonicity of Perelman's $\mu$-functional, but the assertion remains open.