The zero-defect criterion for geodesically harmonic metrics

Let $(M,g)$ be a connected compact Riemannian manifold, and define $\mathcal D(g)$ as the infimum of the defect $\delta(g^\bullet)$ over Riemannian metrics $g^\bullet$ on $SM$ of volume $1$. Define $\mathcal D_{\mathrm{inv}}(g)$ similarly, restricting to $v^g$-invariant metrics. Zero-defect conjecture. If

$$\mathcal D(g)=0,$$

then $g$ is geodesically harmonic; and if

$$\mathcal D_{\mathrm{inv}}(g)=0,$$

then $g$ is invariantly geodesically harmonic. This would characterize the corresponding harmonicity properties by vanishing variational defect; the source gives no resolution.