Some people confuse moral objectivity and moral accountability.

To say that moral values are objective is just to say that moral values are not determined by what anyone thinks. Moral accountability, on the other hand, has to do with the consequences for people’s moral or immoral behavior.

So saying that moral values are objective is to say precisely nothing about moral accountability, including whether there are “eternal consequences” for our actions. It could be the case that there are OMVs and our actions have eternal consequences. It could also be the case that there are OMVs and our actions do NOT have eternal consequences. The existence of OMVs doesn’t make either of these possibilities more likely than the other.