I wouldn't say it's data independent. Look at goldbach's conjecture. While there is no rigorous proof, it seems almost impossible to be the case that it is not true. A lot of conjectures like that are based on data gathered first.
You are referring to an as yet unsolved problem. The maths is as yet un-done!
Sure, there is data that suggests it is probably true. The scientific can say there is enough data to be almost certain it is true and move on. The mathematical community cannot say for certain it is true because it is not yet prooven. Which is why they are still working on it.
I would suggest that formulating conjectures based on data is specifically not maths.
Some mathematicians start with data, and no doubt about it data is effective. But I'm specifically saying that that is employing a tool of the scientific community (see [1]) as a starting point before then proceeding to do some actual mathematical work. This distinction is why maths is often classified in the Arts rather than the sciences.
But if you can point to a maths textbook that teaches someone "general theories" by printing 10,000 data points followed by a QED then I'd suggest it is a pretty extraordinary theory.
EDIT I'll throw in an example; a software consultant might be involved in invoicing for a project. The invoicing is still accounting work, even if it is being done for a software project.
Formulating a conjecture and proving a conjecture are two different activities. A conjecture is based on incomplete information (data gathered so far), and then we try to prove or disprove it using the mathematical tools available (and sometimes developing new tools).
Consider making the observation (shown as a table):
We can come up with the formulation: sum(1 to n) = n(n+1)/2 by several methods, but from the data given it's only a conjecture. Depending on how we came up with that formulation we may already have proven the conjecture. Or if we constructed it using the data only, we can prove it via induction or other methods.
The same is how many other conjectures begin on less trivial examples. The four color theorem, for instance, was notably hard to prove (and there was a lot of controversy over its method of proof). But it was still just a conjecture until they laid out their proof, though no one had ever found a counterexample.
I don't think any mathematician will agree with you on that (I'm a phd student in math). Examples (i.e. data) is foundational to a mathematical intuition which is a foundation of writing proofs. Drawing a line between the two would be ridiculous.
Data for mathematicians rarely looks like tables of numbers. More often it looks like a list of simple manifolds where we can do computations by hand, or topological spaces that don't have the usual properties, or fields of characteristic different from what you're most comfortable with, or continuous functions whose derivative is zero almost everywhere but are not constant. The first thing my advisor asks me when I say "I might be able to prove X" is whether it's true in the simplest examples.
