The conjectured strict exclusion bounds for the cube-approximation function

Let $g$ be the function defined in Theorem, and let $k$ and $i$ be the parameters occurring there. Strict exclusion conjecture. One has

$$g\neq L\quad\text{if }i\leq k-1,$$

and

$$g\neq W\quad\text{if }i\geq k+1.$$

The paper presents this as a stronger result than the bounds proved immediately beforehand; no proof or resolution is given in the supplied text.