After some additional reading and thinking, let me present a more detailed discussion.
In the classical theory of light, described by classical Maxwell equations, interference happens due to the wavy nature of light. More specifically, dark regions of interference correspond to points where the electric field ##E(x)## and magnetic field ##B(x)## vanish.
In the first quantized theory of particles, such as electrons, the field ##\psi(x)## is no longer interpreted as a classical field, but as a probability amplitude. The probability of finding the particle at position ##x## is proportional to ##\psi^∗(x)\psi(x)##.
Can such a first quantized theory be applied to photons? Strictly speaking it cannot, photon must be described by a second quantized formalism. But for the sake of intuition let us cheat a bit and sketch how a first quantized theory of photon would look like. The electric field is real, so can be written as
$$E(x)=\psi(x)+\psi^∗(x)$$
where ##\psi(x)=E^+(x)## is complex. The ##E^+## denotes that its time-dependence involves only positive frequencies, i.e. oscillatory functions of the form ##e^{-i\omega t}##. Then in the first quantized theory, the probability of finding the photon at position ##x## would be proportional to
$$\psi^∗(x)\psi(x)=E^-(x)E^+(x)$$
In reality, the photons are described by the second quantized theory, i.e. quantum optics (which is a part of quantum electrodynamics). In this theory, strictly speaking, we cannot talk of probability to find the photon at position ##x##. But we can talk of probability that a detector at position ##x## will be excited. For a certain model of detector, under certain approximations, the probability of detector excitement at position ##x## is proportional to
$$\langle\Psi| \hat{E}^-(x)\hat{E}^+(x) |\Psi\rangle$$
This looks very similar to the first quantized expression above. However, now ##\hat{E}^+(x)## is an operator (essentially a photon destruction operator), while ## |\Psi\rangle## is a second quantized state of photons. Such a state can contain one photon, or two photons, or any number of photons, or even a state with uncertain number of photons.
When ##|\Psi\rangle## is a one-photon state, one obtains essentially the same result as one would expect from a first quantized theory. However, for some more complicated states, the results may differ from expectations based on first quantized theory.
Even more interesting is the deviation form classical theory. In the classical theory, if ##E^+(x)=0## for some ##x##, then also ##E^-(x)E^+(x)=0##. In the second quantized theory, by contrast, it can be the case that ##\langle\Psi| \hat{E}^+(x) |\Psi\rangle=0##, but
$$\langle\Psi| \hat{E}^-(x)\hat{E}^+(x) |\Psi\rangle \neq 0$$
In other words, the average field may be zero at some point, but the probability of detection may still be non-zero. This suggests that dark regions of interference may have a pattern very different from the classical one.
Now we can finally get to the result of this paper. Essentially, this paper constructs explicit states ##|\Psi\rangle## for which the suggestion above realizes, i.e., for which the second quantized dark regions of interference significantly differ from the classical ones. This explicitly demonstrates that interference in quantum optics cannot be understood as interference of classical waves. We knew it before, but this paper demonstrates it more explicitly. That's interesting, but not revolutionary.
At the end, let me give two references that I used. Instead of the original paper published in PRL, I have used the arXiv version
https://arxiv.org/abs/2112.05512v3
For basics of quantum optics, including the theory of detection, I have used
L.E. Ballentine, Quantum Mechanics A Modern Development, Chapter 19
