Force an object to to be indexed by its id on parent dictionary

Question

I have item objects containing an id property, and a dictionary of said items. I want to type it such that each item must be indexed by its id property on the parent dictionary.

I expected this to do the job:

type Item<K extends string = string> = { id: K }
type Dictionary = { [K in string]: Item<K> }

const dict: Dictionary = { abc: { id: 'xyz' } }

Playground link here

I expected this to result in an error, since "abc" !== "xyz", but TypeScript doesn't seem to care. How come? And how do I achieve my goal?