If it is proven that a given asymmetric encryption protocol relies on a problem which cannot be solved efficiently even by a quantum computer, then quantum cryptography becomes largely irrelevant.

The point is that, as of today, no one was able to do this.
Indeed, such a result would be a serious breakthrough, as it would prove the existence of $\text{NP}$ problems which are not efficiently solvable on a quantum computer
(while this is generally believed to be the case, it is still unknown whether there are problems in $\text{NP}\!\setminus\!\text{BQP}$).

Generally speaking, all classical asymmetric encryption protocols are safe under the assumption that a given problem is hard to solve, but in no case, to my knowledge, it has been proven (in the computational complexity sense) that that problem is indeed exponentially hard to solve with a quantum computer (and for many not even that the problem is not efficiently solvable with a *classical* computer).

I think this is nicely explained by Bernstein in his review of post-quantum cryptography (Link).
Quoting from the first section in the above, where he just talked about a number of classical encryption protocols:

Is there a better attack on these systems? Perhaps. This is a familiar
risk in cryptography. This is why the community invests huge amounts
of time and energy in cryptanalysis. Sometimes cryptanalysts find a
devastating attack, demonstrating that a system is useless for
cryptography; for example, every usable choice of parameters for the
Merkle–Hellman knapsack public-key encryption system is easily
breakable. Sometimes cryptanalysts find attacks that are not so
devastating but that force larger key sizes. Sometimes cryptanalysts
study systems for years without finding any improved attacks, and the
cryptographic community begins to build confidence that the best
possible attack has been found—or at least that real-world attackers
will not be able to come up with anything better.

On the other hand, the security of QKD does, *ideally*, not rely on conjectures (or, as it is often put, QKD protocols provide in principle *information-theoretic security*). If the two parties share a secure key, then the communication channel is unconditionally secure, and QKD provides an unconditionally secure way for them to exchange such a key (of course, still under the assumption of quantum mechanics being right).
In Section 4 of the above-mentioned review, the author presents a direct (if possibly somewhat biased) comparison of QKD vs Post-Quantum cryptography.
It is important to note that of course "unconditional security" is here to be meant in the information-theoretic sense, while in the real world there may be more important security aspects to consider.
It is also to be noted that the real-world security and practicality of QKD is not believed to be factual by some (see e.g. Bernstein here and the related discussion on QKD on crypto.SE), and that the information-theoretic security of QKD protocols is only true if they are properly followed, which in particular means that the shared key has to be used as a one-time pad.

Finally, in reality, also many QKD protocols may be broken. The reason is that experimental imperfection of specific implementations can be exploited to break the protocol (see e.g. 1505.05303, and pag.6 of npjqi201625).
It is still possible to ensure the security against such attacks using device-independent QKD protocols, whose security relies on Bell's inequalities violations and can be proven to not depend on the implementation details.
The catch is that these protocols are even harder to implement than regular QKD.

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– Sanchayan Dutta – 2019-05-15T14:16:12.093