Uniform volume doubling for left-invariant metrics on compact Lie groups

Let $K$ be a connected real compact Lie group, and let $\mathfrak{L}(K)$ denote the family of all left-invariant Riemannian metrics $g$ on $K$. For each $g\in\mathfrak{L}(K)$, let $d_g$ be the Riemannian distance, $\mu_g$ the Riemannian volume measure, and

$$D_g:=\sup_{x\in K,\,r>0}\frac{\mu_g(B_{d_g}(x,2r))}{\mu_g(B_{d_g}(x,r))}$$

be the volume-doubling constant.

Uniform volume-doubling conjecture. There is a constant $D(K)$ such that

$$D_g\leqslant D(K)\qquad\text{for all }g\in\mathfrak{L}(K),$$

that is, $K$ is uniformly doubling with constant $D(K)$.

Uniform volume doubling is presented as a simpler conjectural question implying the two-sided spectral and heat-kernel bounds above in the compact connected Lie-group setting. The paper proves the result for left-invariant geometries on $\operatorname{SU}(2)$, while the general compact-Lie-group statement remains open in the supplied text.