Formalize the Hopf problem: does the 6-sphere admit a complex structure?

### What is the conjecture

The **Hopf problem** (Hopf, 1948): *does `S⁶` admit a complex structure?* One of the oldest open problems of complex geometry. `S²` and `S⁶` are the only spheres admitting almost complex structures (Borel–Serre), and on `S⁶` neither the integrability of some almost complex structure nor its impossibility has been established. LeBrun (1987) showed no complex structure on `S⁶` is orthogonal for the round metric. Claimed resolutions in both directions have not achieved community acceptance.

Reference: I. Agricola, G. Bazzoni, O. Goertsches, P. Konstantis, S. Rollenske, [*On the history of the Hopf problem*](https://arxiv.org/abs/1708.01068), Differ. Geom. Appl. 57 (2018).

### Prerequisites needed

None beyond current Mathlib: the statement is expressible today as the existence of a `ChartedSpace (EuclideanSpace ℂ (Fin 3))` structure on `Metric.sphere (0 : EuclideanSpace ℝ (Fin 7)) 1` satisfying `IsManifold 𝓘(ℂ, EuclideanSpace ℂ (Fin 3)) ω`. Since the answer is unknown in both directions, the `answer(sorry) ↔` form applies.

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

* ams-32
* ams-53

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

The issue provides no file or test entry point. Start by locating the repository's conventions for formalized conjectures and the existing definitions for ChartedSpace and IsManifold, then express the stated S⁶ complex-structure existence claim using the indicated answer(sorry) ↔ form. Done means the conjecture is added in the repository's accepted Lean format.

Written by the indexing model from the issue text.