Hitting time conjecture for connectivity in the W-edge-incremental process

Let $W$ be a connected graphon and $K\in\mathbb{N}$. For each $n\in\mathbb{N}$, consider the $W$-edge-incremental process $\mathfrak{X}_n=(G_n^t)_{t=0}^{\binom{n}{2}}$ of order $n$. Let $f_n$ and $g_n$ be the hitting times in this process for the properties of being $K$-connected and having minimum degree $K$, respectively. Hitting time conjecture. Asymptotically almost surely,

$$f_n=g_n.$$

This extends the Erdős–Rényi hitting-time result to the inhomogeneous setting and to higher minimum degree and connectivity. The conjecture concerns the edge-incremental process associated with a connected graphon.