The multi-arc smoothing lower-bound conjecture

Let $calpha$ be a multi-arc with $n$ arc components and $k e 1$ self-intersections. Suppose that $calpha$ is taut with free endpoints. A smoothing is a smoothing of $calpha$ as defined in the paper.

Multi-arc smoothing lower-bound conjecture. The multi-arc $calpha$ has a smoothing that cannot be homotoped to a multi-arc with fewer than

$$k-(n+1)$$

self-intersections.

This conjecture seeks a quantitative lower bound for the complexity remaining after smoothing a taut multi-arc with free endpoints. The source presents it as expected rather than proved, and no resolution is supplied.