I've heard about this experiment, but I haven't studied it. Nor am I as smart as Leonard Mandel. With those declaimers I say: I interpret this result differently. (If a lot of people try to correct my errors in thinking, I'll turn this into a separate question.) And I apologize for not actually answering the question. But I think the premise of the question is flawed, and requires comment.
In my view, each laser is exciting the same mode. Each is increasing the photon number of the same mode. The shape of the mode, which is the thing that determines what the interference pattern looks like, is determined by the wavefunction of the mode. The wavefunction of the mode is determined by the geometry of the situation, as is true of all wavefunctions. In the case of an EM mode, the wavefunction is provided by solutions of the EM wave equation. And like all wavefunctions, it is always populated by a zero-point excitation. That is, even in the absence of an intentional excitation the mode exists and is occupied.
And as we know, the solution of the wave equation for a two-slit experiment is an interference pattern.
Each "photon" (additional excitation to the EM mode) excites the entire mode. It makes no sense to think of photons traveling through one or the other slit. The mode, and the excitation, exists in both slits. When it gets to the detector/screen there is wavefunction amplitude only where the interference pattern is not zero. In accordance with the usual interpretation of wavefunction, there is some probability for the interaction between the mode and the detector in proportion to the squared amplitude of the wavefunction. Detections occur only occur where the interference pattern is non-zero, and when the interaction occurs discrete quantities of energy and momentum are transferred from the mode to the detector/screen, and looks for all the world like a particle hit the screen.
Additionally, in this view interference has nothing to do with "overlapping photons" whatever that might mean. We don't have to have particles that somehow interfere with each other. (I can't imagine what "two particles interfering" might mean.) So it doesn't matter at all that the photon rate in this experiment is slow enough that individual photons never exist at the same time. (What this statement really means is that the rate of discrete detections is low enough that there are no coincident detections. No mention here of "photon particles hitting a detector".)
So I still subscribe to Dirac's statement "Interference between different photons never occurs." Some people reject that statement.
I took a quick look at the Pfleegor-Mandel experiment. They don't measure fringes at all, so a lot of what I say above doesn't apply to their experiment. I'm don't fully understand their experiment yet. It appears that they measure intensity correlations between photons from two different sources. This is not the same as a traditional interference experiment which measures amplitude correlations. They find correlations that oscillate in space in the same way one would expect as from a single source and a beam splitter. The importance of the very slow photon rate is that the photons should be generated with enough temporal separation that any phase relationship between the two sources should have diffused away to zero.
With this (very small) improved understanding I will say that I would have to come up with a different interpretation of P-M that matches my picture. My description above does not apply to the P-M experiment. In fact, the P-F experiment is subtle enough that I think it requires a full quantum mechanical analysis to understand. (I note that even then it's possible that two interpretations can apply to the same theory. For example, the theory of shot noise has more than one interpretation.)
Pfleegor and Mandel themselves stop short of a definitive interpretation. Since then a lot of people have thought about it and done follow-up experiments. The people who did them probably have a good understanding of things. Nonetheless, none of these experiments involve traditional two-slit interference and a single detector/screen. Dirac was certainly not thinking of higher-order correlations when he made his statement, and we (which includes me) shouldn't be applying naive understanding of two-slit interference when trying to interpret these experiments. What Dirac said is probably true as far as it goes, but it may not go far enough.