Establish the conjectured lower bounds for multiplicative spanner construction

Establish the conjectured quantum query lower bound of \(\Omega(n^{1+2/k})\) for even stretch \(k\), or \(\Omega(n^{1+2/(k+1)})\) for odd stretch \(k\), for explicit multiplicative \(k\)-spanner construction beyond the currently resolved stretch values.

Background

The output-length method for spanner lower bounds depends on the existence of sufficiently dense high-girth graphs. The paper notes that known extremal constructions establish only selected stretch cases, while the conjectured lower-bound exponents would apply more broadly. The first missing case is $k\in\{7,8\}$, and the conjectured lower bound remains open for every stretch beyond ten; the paper obtains an unconditional $\Omega(n^{5/4})$ bound for $k=7,8$ by a different reduction.