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on Minimum spanning-tree dilation on the regular nonagon

The statement

Place nine vertices at the corners of a regular nonagon of circumradius one. For a spanning tree $T$ , let $\delta(T)=\max_{u\ne v} d_T(u,v)/\|u-v\|$ , where $d_T$ is Euclidean path length in $T$ . Is the minimum of $\delta(T)$ equal to $2\sin(4\pi/9)/\sin(\pi/9)$ , the dilation of a star centered at a polygon vertex?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.