The burden of proof is on the person claiming something is true.
The burden of proof is on the person making an assertion. The original claim was not an assertion, it was that they "may be slightly conscious". The article linked here is the only one that made an actual assertion, which is that the original claim was categorically false.
In short, I agree with you. The burden of proof is on this article to demonstrate that neural nets are not conscious.
First: What is consciousness? That's the first thing we need to do. Come up with a definition that we can both agree on absent any particular example. So we can't hold up a human and just shrug and gesture in its direction.
Now. We need to prove that humans fit that bill. And in a way that precludes other philosophically possible scenarios. Or at least agree on some fundamental axioms. Like assuming that no one is a brain in a vat. But if we're trying to prove humans are conscious and not simply automatons, can we even dismiss solipsism? Because if humans are not conscious, then I could be a brain in a vat and they're just automatons running for a separate reason.
And the truth is, we can't get past even those two hurdles. We assume humans are conscious because we kind of define consciousness as just shrugging and gesturing in its general direction.
Even if we call consciousness "awareness of self", how do we prove something is aware of itself and not just making the claim without understanding.