For all we know, it is extraordinarily hard to prove that a problem which can be solved by a quantum computer is classically hard.
The reason is that this would solve an important and long-standing open problem in complexity theory, namely whether PSPACE is larger than P.
Specifically, any problem which can be solved by a quantum computer in polynomial time can also be solved in polynomial space by a classical computer (the class PSPACE). However, it is not known whether PSPACE is strictly larger than P (the class of problems efficiently solvable on a classical computer). This is a long-standing open question in complexity theory, and thus any hardness result as the ones you talk about would also resolve that question, making it an extremely hard problem.
(In fact, there are tighter upper bounds on BQP, most importantly the class PP, but separating PP from P is even harder.)
This might be a reason for Preskill's careful formulation
Quantum supremacy or quantum advantage is the potential ability of quantum computing devices to solve problems that classical computers practically cannot.
It should however be said that this does not mean that we cannot make any statement about the hardness of certain problems: What we can do is to relate their hardness to the hardness of other problems, for which -- while equally not proven -- there is a lot of accumulated evidence: For instance, it might be known that if problem X is easy, a range of other problems would also be easy, which have long resisted solution.
Thus, while we cannot unconditionally prove hardness of a problem, it is possible to relate it to the hardness of other problems whose hardness, while also unproven, are far better backed up with evidence.