The approximability frontier conjecture for sparse Diameter

Let sparse graphs have $m=\tilde{O}(n)$ edges, and consider the four variants obtained by choosing directed or undirected graphs and weighted or unweighted edges. For $k\in\mathbb N^+\cup\{\infty\}$, approximation within a ratio means returning a value within that multiplicative factor of the sparse Diameter. The approximability frontier conjecture. Sparse directed or undirected, weighted or unweighted Diameter is $(2-1/k)$-approximable in time $\tilde{O}(n^{(k+1)/k})$ for every $k\in\mathbb N^+\cup\{\infty\}$. Unless SETH fails, approximating sparse directed or undirected, weighted or unweighted Diameter within a ratio better than $2-1/(k+1)$ requires time $n^{(k+1)/k-o(1)}$ for every $k\in\mathbb N^+$. This predicts that the four sparse Diameter variants share the same approximability frontier, matching the paper's proposed unifying landscape; the conditional lower-bound part depends on SETH and remains conjectural.