Babai spectrum conjecture for paths

Let $n$ and $k$ be fixed positive integers with

$$\left\lfloor \frac{n}{2} \right\rfloor < k \leq n-1,$$

and let $m\in\{2,3,\ldots,n-1\}$ be the unique integer satisfying

$$n-\left\lceil \frac{n}{m} \right\rceil < k \leq n-\left\lceil \frac{n}{m+1} \right\rceil.$$

For the path $P_n$ and its $k$-Babai spectrum $\operatorname{Spec}(P_n,k)$, the preceding theorem gives

$$\{m+1,m+2,\ldots,k+1\}\subseteq \operatorname{Spec}(P_n,k).$$

Babai spectrum conjecture for paths. Under these hypotheses,

$$\operatorname{Spec}(P_n,k)=\{m+1,m+2,\ldots,k+1\}.$$

The conjecture strengthens the preceding theorem's inclusion to an exact description of the Babai spectrum. The authors say that numerical examples motivate it, but no resolution is supplied in the source.