Hamilton powers conjecture for inhomogeneous random graphs

Let $W:\Omega^2\to[0,1]$ be a graphon, let $k\ge 2$ be fixed, and let $\mathbb{G}(n,W)$ be the associated inhomogeneous random graph. A $K_{k+1}$-fractional cover of $W$ is a measurable function $f:\Omega\to[0,1]$ such that for almost every $(k+1)$-tuple $(x_1,\ldots,x_{k+1})$, either $\sum_{i=1}^{k+1}f(x_i)\ge1$ or at least one pair among the tuple has graphon value zero. Hamilton-power conjecture. The graph $\mathbb{G}(n,W)$ contains asymptotically almost surely the $k$-th power of a Hamilton cycle if and only if: $W$ is a connected graphon; $\lim_{\alpha\searrow0}\mu(\mathsf{D}_\alpha(W))/\alpha=0$; and the only $K_{k+1}$-fractional cover of $W$ with $L^1$-norm at most $1/(k+1)$ is the constant function $1/(k+1)$. The conjecture extends the Hamilton-cycle characterization to higher powers, replacing the perfect-matching obstruction by the corresponding $K_{k+1}$-tiling obstruction.