Inclusion modulo nonstationary

A classical theorem of Hechler asserts that the structure (ω^ω, ≤ ^∗) is universal in the sense that for any σ-directed poset P with no maximal element, there is a ccc forcing extension in which (ω^ω, ≤ ^∗) contains a cofinal order-isomorphic copy of P. In this paper, we prove the following consistency result concerning the universality of the higher analogue (κ^κ, ≤ ^S) : assuming GCH, for every regular uncountable cardinal κ, there is a cofinality-preserving GCH-preserving forcing extension in which for every analytic quasi-order Q over κ^κ and every stationary subset S of κ, there is a Lipschitz map reducing Q to (κ^κ, ≤ ^S).