Inclusion modulo nonstationary

A classical theorem of Hechler asserts that the structure $\left(\omega^\omega,\le^*\right)$ is universal in the sense that for any $\sigma$-directed poset P with no maximal element, there is a ccc forcing extension in which $\left(\omega^\omega,\le^*\right)$ contains a cofinal order-isomorphic copy of P. In this paper, we prove a consistency result concerning the universality of the higher analogue $\left(\kappa^\kappa,\le^S\right)$: Theorem. Assume GCH. For every regular uncountable cardinal $\kappa$, there is a cofinality-preserving GCH-preserving forcing extension in which for every analytic quasi-order Q over $\kappa^\kappa$ and every stationary subset S of $\kappa$, there is a Lipschitz map reducing Q to $(\kappa^\kappa,\le^S)$.