Asymptotic extremal-edge conjecture for convex-packable plane paths

Let $f(k)$ be the maximum number of extremal edges of a convex-packable plane path with $k$ edges. The source establishes the upper bound $f(k)\leq 2\sqrt{k}$ for every positive integer $k$. Extremal-edge conjecture.

$$f(k)=(2-o(1))\sqrt{k}\quad\text{as }k\to\infty.$$

This conjecture asserts that the upper bound is asymptotically sharp for convex-packable plane paths. It concerns the asymptotic structure of convex geometric packings; the source gives the upper bound but leaves sharpness as a conjecture.