Reverse comparison between residual matching number and matching variance

Determine whether there is an absolute constant C such that the residual matching number λ(G) is at most C times the matching variance σ²(G) for every finite simple graph G; equivalently, establish whether λ=O(σ²).

Background

The paper disproves the bound σ²=O(λ) by constructing graphs for which σ²/λ tends to infinity, thereby refuting λ=Θ(σ²). It explicitly leaves the reverse inequality λ=O(σ²) unresolved.