Moral supervenience (S) is the thesis that the moral truths supervene on the "natural" (read: "non-normative") truths. This seems self-evident: moral facts cannot "float free" of the underlying natural facts -- there could not be a world just like ours, but where Hitler was actually a virtuous fellow. Such a radical moral difference would need to be grounded in a corresponding natural difference in the world -- say where Adolf never become a fascist, and instead dedicated himself to the peaceful pursuit of philanthropic ends. To expand upon this intuitive point:
(1) There seems an important sense in which the descriptive facts are what ground the particular moral facts. But without moral supervenience, the descriptive facts would be insufficient to determine what moral facts hold in a situation. There would instead be brute moral truths that aren't fully grounded by the descriptive facts, insofar as the former could change despite no change in the latter. Not only is this intuitively crazy, but it also violates our theoretical sense of how the particular moral truths are grounded.
(2) The fundamental moral truths are plausibly necessarily true conditional statements, perhaps of the form, "In completely described natural circumstances Ci, the applicable moral truths are Mi." This then explains the sense in which the particular moral truths Mi are grounded in the descriptive truths Ci. But it also straightforwardly entails moral supervenience (S).
(3) The falsity of S would make nonsense of the guiding role of moral truths. Note that, e.g., if someone is virtuous, or morally admirable, then it's fitting for us to admire them, or hold them in a certain positive regard. Moreover, to avoid moral fetishism, what we should admire in them is not "virtue" in the abstract, but rather the particular psychological characteristics that are virtues. But if S is false, then our emotions and attitudes cannot be guided in this way. It makes no sense to admire someone for their psychological characteristics in our world, but then to condemn them for being exactly the same way in exactly the same descriptive circumstances in another world, where the moral truths about them brutely differ.
Harrison addresses something along these lines:
The principle of universalizability (U) states that if an action is right, then every action that is similar in morally relevant respects is right too.
However, he holds that the denial of S is compatible with U, because a brute moral difference would, trivially, be a difference in "morally relevant respects".
Of course, U is generally not meant to be interpreted so trivially. Rather, when we speak of "morally relevant respects" here, we mean the underlying descriptive properties that have the higher-order property of being morally important. Harrison goes on to consider:
[U2] if act x is right, then any act having the same properties in virtue of which act x is right, is also right.
Unfortunately, he reads U2 as entailing an implausible absolutism:
Act A might be wrong because it was an act of cruelty. Act B might also have been an act of cruelty in exactly the same way, yet have saved a million lives. According to U2 if A is wrong, B is too.
This, too, is not the best interpretation of Universalizability. We should not be considering right-makers and wrong-makers in isolation, but rather the totality of natural facts that together entail the rightness or wrongness of an act. This may include "negative" facts, such as that act A does not save a million lives. That's a morally relevant natural feature of A, which differs from B, and explains why one may consistently reach different moral verdicts about the two.
When U2 is understood in this (obviously more sensible!) way, it is far harder to deny. The whole paper seems to rest on neglect of this crucial point. (It re-emerges later when Harrison takes a grounding-style claim to be motivated by U2.)
Admittedly, if one is determined to reject the self-evidence of S, and hence even the better interpretation of Universalizability that I've offered here, then I'm not sure what more could be said to persuade them to reconsider. But it would seem awfully odd and unmotivated.