Sum-diameter conjecture for paths

Let $P_n$ denote the path graph on $n$ vertices. The sum-diameter of a graph $G$, denoted $\operatorname{sd}(G)$, is the minimum range of a labeling whose induced sum graph is $G$.

Sum-diameter conjecture for paths. For $n\geq 3$, we have

$$\operatorname{sd}(P_n)=\begin{cases} 2n-3 & 3\leq n\leq 6,\\ 2n-2 & n\geq 7. \end{cases}$$

An exhaustive computer search verifies the values through $n=13$, but the asserted formula beyond the computed range remains open. The conjecture refines the preceding bounds for the sum-diameter of paths.