Distance-realization property for geodesics in the Heisenberg group

Let $S_R$ denote the metric sphere of radius $R$ in the Heisenberg group, and let $P\in S_1$. Suppose that the geodesic leaving the origin and reaching $P$ meets the $t$-axis before it meets $S_R$.

Distance-realization property. Then the distance $\operatorname{dist}(P,S_R)$ is realized by the point

$$\left(0,\frac{R^2}{\pi}\right).$$

The statement proposes a specific nearest point on the larger sphere under the stated geodesic condition. The supplied text presents it as a property to study and gives no evidence that it has been proved or disproved.