The maximal-distance conjecture for tangent cones of harmonic maps

Let $Y$ be the object under consideration, let $p$ be a point, and let $TC_pY$ denote its tangent cone at $p$. Write $0_p$ for the zero element in this tangent cone, and let $\delta$ denote the distance between the indicated objects. The maximal-distance conjecture.

$$\delta(TC_pY,0_p)=\frac{(\sqrt{r}-1)^2}{2(r-\sqrt{r}+1)}.$$

This conjecture is presented as a partial confirmation following an estimate that establishes the corresponding upper bound except when $r=2$; the supplied text does not state whether the equality is known in general.