I described very carefully in my previous answer the two possible conceptualisations of the OP's example, explaining why both singular and plural agreement are possible.
The agreement patterns in a neither nor coordination are:
1. neither sg nor sg = sg
2. neither sg nor pl = pl
3. neither pl nor sg = pl
4. neither pl nor pl = pl
And the patterns in an either or coordination are:
5. either sg or sg = sg
6. either sg or pl = sg/pl (many speakers would avoid this coordination altogether)
7. either pl or sg = sg/pl (many speakers would avoid this coordination altogether)
8. either pl or pl = pl