Necessity of logarithmic factors in width comparisons

Determine whether the bounds $fhw(H)=O(\lambda(H)\,adw(H))$ for all hypergraphs and $ghw(H)=O(adw(H))$ for hypergraphs with private intersections hold, or alternatively establish matching logarithmic separations showing that the logarithmic factors in the known comparisons are necessary.

Background

The paper proves an $O(\lambda(H)adw(H)\log adw(H))$ upper bound for fractional hypertree width and cites a complementary $O(adw(H)\log adw(H))$ bound for generalised hypertree width under private intersections. The logarithmic factors arise from different technical steps in the separator and flow arguments. No known example demonstrates that either logarithmic loss is necessary, leaving unresolved whether both comparisons can be improved to remove the logarithm or whether matching lower-bound separations exist.