Spum conjecture for paths

Let $P_n$ denote the path graph on $n$ vertices, and let $\rho(L)=\max L-\min L$ be the range of a labeling $L$. The spum of $P_n$, denoted $\operatorname{spum}(P_n)$, is the minimum range among labelings that induce $P_n$ as a sum graph.

Spum conjecture for paths. For $n\geq 8$, we have

$$\operatorname{spum}(P_n)=\begin{cases} 2n+1 & \text{if } n \text{ is odd},\\ 2n-1 & \text{if } n \text{ is even}. \end{cases}$$

The paper notes that the even case improves and falsifies a previous conjecture, while exhaustive computation verifies the claimed pattern only for the finite range discussed in the surrounding text; the general assertion remains open.