Intel's L3 cache is inclusive. Spectre most definitely also applies across different cores.
If privileged system code never runs on the same physical core(s) as user code, and so we leave out branch prediction leaks, we are dealing only with cache timing leaks via L3 cache (Meltdown attack). But in the data segment of pure microkernel (which only does IPC and task switching) there's not much to hunt for.