You cannot falsify mathematics by experiment (except in the subjective Bayesian sense).
Actually, that's technically false. The statements mathematical axioms make about reality are bizarre, but they exist and are actually falsifiable.
One of the fundamental properties we want from our axiomatic systems is consistency — the fact that it does not lead to a logical contradiction. We would certainly reject our current axiomatic foundations in case we found them inconsistent.
Turns out it's possible to write a program which would halt if and only if ZFC is consistent. I would not recommend running this one as it's a Turing machine and thus not really optimized (and in any case, ZFC being inconsistent is unlikely, and it's even more unlikely that the proof of it's inconsistency would be easy to be found with current technology), but in theory one might run one of such machines long enough to produce a contradiction, which would basically physically falsify the axioms.
It sounds like this is making a case for a view known in the philosophical literature as moral response-dependence. PhilPapers offers 39 papers on this topic; here is one counterpoint that I recommend.
I have to say that the claimed reductios here strike me as under-argued, particularly when there are literally decades of arguments articulating and defending various versions of moral anti-realism, and which set out a range of ways in which the implications, though decidedly troubling, need not be absurd.
There are three branches of philosophy: natural philosophy, moral philosophy and metaphysical philosophy. Colloquially, we refer to natural philosophy as "science", "moral philosophy as "ethics" and metaphysical philosophy as "philosophy". There's also math "in fact, it would not be a bad definition of math to call it the study of terms that have precise meanings."
Science is a worthwhile endeavor. Metaphysical philosophy tends to be worthless because it makes no falsifiable predictions about reality.
What about ethics? Western philosophy tends to take it for granted that ethics is consequentialist. Consequentialist ethics breaks ethics down into three components.
Modelling the world is science. Optimizing a value function is math. Science and math are ethically trivial because consequentialist value functions are orthogonal to science and math. Value functions can be anything, which is why consequentialist moral paradigms tend to produce paperclip maximizers.
Consequentialist moral frameworks reduce ethics to axiomatic value functions (which could be anything because they are are unfalsifiable) and optimization strategies (which are ethically trivial). The optimization strategies belong in the realms of science and math. The axiomatic value functions of a consequentialist moral framework are declared by fiat. They are upstream of reality, like mathematical axioms.
It is fine for mathematical axioms to be upstream of reality because mathematical axioms don't make statements about reality. The Axiom of Choice isn't a statement about reality. It's a statement about Mathland. The Axiom of Choice isn't even a statement about all of Mathland. The Axiom of Choice is applies only to the region of Mathland where the Axiom of Choice applies. Two plus two equals four in the real numbers R but two plus two equals zero in the Z4 cyclic group.
It's fine for mathematical axioms to be unfalsifiable because math itself is upstream of reality. You cannot falsify mathematics by experiment (except in the subjective Bayesian sense).
Ethics must be falsifiable because of reducto ad absurdum. If ethics wasn't falsifiable then there would be no truth values to statements like "genocide is bad", "compassion is good" or even "suffering is bad". Could you build an AI with morality inverted from human beings? Yes. But human morality is broadly consistent. Syrian dictator Bashar Assad's family reads Harry Potter. Vladimir Putin understands the value of loyalty.
Human morality isn't perfectly consistent. Ethics is like taste.
If you throw out the idea that ethical axioms can be right or wrong then you must throw out the idea you can make good and bad choices at all which is another reducto ad absurdum. There might not be such thing as good and bad choices in the objective reductionist physical sense but they are a very real part of subjective human experience.
Ethics, like art, has a human audience.