From [Holomorphic 2-vector bundles on nonalgebraic 2-tori, Paul Flondor / Vasile Brinzanescu, J. reine angew. Math, 1985]: "It is known (see Schwarzenberger) that on a projective surface every topological 2-vector bundle has a holomorphic structure iff its first Chern class is of the form $c_1(L)$ with $L$ a holomorphic line bundle."

Thank you. This is an interesting work but it is related to a ``good'' part of the Hawking's parer (the one I have no issues with) about the area theorem. It does not address the question I am asking.

My (not very educated) opinion is that the conjecture is more likely to be true then false. And, my wild guess is that $l^{100}$ or at least $l^{1000}$ should be enough to see the number unless the variety is not very nice. Of course, a much more interesting possibility is disproving the conjecture (and collecting $1M). But yes, one cannot do it this way having no rigorous bound on the height, and obtaining the latter may be rather difficult.

Technically yes. But the transition to the general case is more or less straightforward, so apparently nobody cared much about the difference.

What if $X$ is compact - say, a sphere? You obviously need some more conditions (e.g. negative curvature) for the question to make sense.

I agree, but when mathematics and physics were the same constructive logic did not exist (and for a reason, I think). It may be that we mean different things when speaking of a model. Can you give an example when it matters if the foundations a particular model is built on are classical or constructive?