GMAT Sentence Correction (SC) | The usage of OR in sentence correction

Dear Expert,

Does the connecting word "OR" follow the same rule as "Either ... Or" and "Neither ... Nor"?
THAT IS If one noun is singular and the other is plural, find the noun nearest to the verb, make sure that the verb agrees in number with this noun.

eg.
Mary or Sandy IS/ ARE
Cody, Mary or Sandy IS/ ARE
Cakes (plural) or rice (singular) IS / ARE
Rice or Cakes IS / ARE

Please clarify.

Thanks.

---