Spectral Sets, Rationality of the Spectrum

### What is the conjecture

Let $E \subset \mathbb{R}$ be a bounded spectral set. Is its spectrum $\Lambda$ necessarily rational? More generally, does every spectral $E$ admit some rational spectrum?

* Question 5 in https://arxiv.org/pdf/2408.15361
* For definition of spectral and spectrum, see https://en.wikipedia.org/wiki/Fuglede%27s_conjecture

### Prerequisites needed
Mathematical context that needs to be formalized so that this conjecture can be added.
Consider linking any open Mathlib issues / PRs here.

**Depends on** #3497 (Fuglede's conjecture, helper definitions/lemmas for spectral sets in FormalConjecturesForMathlib)

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

* ams-42
* ams-46

### Choose either option
- [ ] I plan on adding this conjecture to the repository
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

Start by reading the helper definitions and lemmas planned in #3497, then consult Question 5 in the linked paper and the referenced definition of spectral sets. The work is done when the conjecture and its required mathematical context are formalized in the repository.

Written by the indexing model from the issue text.