Polynomial stabilization conjecture for iterative convex-hull construction

Let $P$ be a finite point set in a $2$D CAT(0) complex with a single vertex $O$. Set $H_0=P$; for each $i\geq 1$, let $H_i$ be obtained from $H_{i-1}$ by adding all intersections of shortest paths between pairs of points in $H_{i-1}$ with edges of the complex.

Polynomial stabilization conjecture. The iterative construction stabilizes after a polynomially bounded number of iterations:

$$H_k=H_{k+1}$$

for some polynomially bounded $k$.

The preceding discussion establishes that the convex hull is closed, so the procedure is finite; the conjecture asks for a polynomial bound on the number of iterations needed for stabilization.