Asymptotic affine form of successive giant-component thresholds

For each $k\geqslant1$, let $\sigma_k$ be the threshold at which the limiting giant-component function $\rho_k(t)$ becomes positive. Threshold asymptotics conjecture. There exists a real constant $\sigma_\infty$ such that, as $k\to\infty$,

$$\sigma_k=2k+\sigma_\infty+o(1).$$

In particular,

$$\sigma_k-\sigma_{k-1}\to2.$$

This is a consequence suggested by the conjectured translated universal profile, but remains open.