I think Peterson clearly wins this round.
Whatever is wrong with the thinking of theists and flat-earthers and all, very few of them think that their beliefs are justified by their own sincerity. I mean, who does ever think that? The reason to mention sincerity is to rebut the charge of deliberate fraud, not to justify one's belief. Deluded people think they have good reasons for their beliefs, not just that sincere belief is its own justification.
And this should be obvious enough, if you think for a moment, that it seems clear that gemli didn't think for that moment, but just fired off a zinger because he liked how it sounded. He seems to have made a nonsensical assertion, about other people's thinking, just because it feels truth-y to him. Peterson may be entitled to the Hoist With Their Own Petard achievement badge.
And the double slit experiment of 1801 did
not demonstrate the wave-particle duality of light. It demonstrated the wave nature of light, at a time when particle theories of light were widely accepted. People argued back and forth about whether light was waves or particles, but it would be more than a century before anyone even suggested that it could be both. The early interference experiments showed no trace whatever of any particle properties, and were universally understood as demonstrating that light was not a particle at all, but was only a wave. Holdouts for particle theories, if there were any, had to try to explain those experiments away.
A 1908 experiment by G.I. Taylor is often cited as experimental proof of wave-particle duality, but it wasn't. It was relevant, all right: the experiment was intended to investigate a pre-quantum-mechanical version of wave-particle duality, in which the electromagnetic field in a light wave was proposed to form small clumps. All that Taylor's experiment actually showed, though, was that perfectly wave-like interference behaviour persisted even for extremely faint light, with no sign of any lumps or clumps. People who wanted to believe in a particle character of light, for other reasons, found ways to interpret Taylor's perfect wave results in wave-particle in terms. Taylor's own initial conclusion was just that the lumps had to have been too small for him to resolve. After photons and quantum mechanics were accepted, people started saying that Taylor's experiment showed that a single photon could interfere with itself. In fact Taylor's five photographs of faint-light interference stripes were merely consistent with this photon self-interference hypothesis—they did not demonstrate it. Photon self-interference was a way to explain away Taylor's experiment and save the wave-particle theory.
Much later analysis has shown that Taylor's faintest light was still well above the single-photon threshold, so he was actually right, more or less, about the lumps being too small for him to resolve. Really clean, in-your-face demonstrations of wave-particle duality with double-slit experiments have in fact now been made, but the earliest really good one I could find was
from 1981. For one thing the clean experiments use lasers. Well into my own lifetime, it seems, the wave-particle duality of light was a firmly accepted scientific fact that was justified through a theoretical narrative that could connect disparate data, and not as in gemli's model by any single unambiguous observation.
Nowadays, sure, wave-particle duality can meet gemli's epistemological standard. I'm glad that the physicists of the 20th century had weaker standards than that, though, or we might never have believed strongly enough in photons to make all the progress we did.