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Proving only over source code that programs do not leak sensitive data leaves a gap between reasoning and reality that can only be filled by accounting for the behaviour of the compiler. Furthermore, software does not always have the luxury of limiting itself to single-threaded computation with resources statically dedicated to each user to ensure the confidentiality of their data. This results in mixed-sensitivity concurrent programs, which might reuse memory shared between their threads to hold data of different sensitivity levels at different times; for such programs, a compiler must preserve the value-dependent coordination of such mixed-sensitivity reuse despite the impact of concurrency. Here we demonstrate, using Isabelle/HOL, that it is feasible to verify that a compiler preserves noninterference, the strictest kind of confidentiality property, for mixed-sensitivity concurrent programs. First, we present notions of refinement that preserve a concurrent value-dependent notion of noninterference that we have designed to support such programs. As proving noninterference-preserving refinement can be considerably more complex than the standard refinements typically used to verify semantics-preserving compilation, our notions include a decomposition principle that separates the semantics-preservation from security-preservation concerns. Second, we demonstrate that these refinement notions are applicable to verified secure compilation, by exercising them on a single-pass compiler for mixed-sensitivity concurrent programs that synchronise using mutex locks, from a generic imperative language to a generic RISC-style assembly language. Finally, we execute our compiler on a nontrivial mixed-sensitivity concurrent program modelling a real-world use case, thus preserving its source-level noninterference properties down to an assembly-level model automatically. (Full abstract in paper)