Priestley's conjecture on approximation for weighted k-edge-connected spanning multigraphs

Let $G=(V,E)$ be an edge-weighted graph, and let $k$-ECSM denote the problem of finding a minimum-cost $k$-edge-connected spanning multigraph, where edges may be used with arbitrary nonnegative integer multiplicity. An approximation algorithm is polynomial-time if its running time is polynomial in the input size.

Priestley's conjecture. $k$-ECSM admits a polynomial-time $(1+O(1/k))$-approximation algorithm.

For unweighted graphs, a $(1+2/k)$-approximation is known, while the conjectured asymptotic dependence on $k$ for general edge weights was posed as an open question.