The general parsing rules state:

• if that child element itself has a microformat ("h-*" or backcompat roots) and is a property element, add it into the array of values for that property as a { } structure, add to that { } structure:
  • value:
    • if it's a p-* property element, use the first p-name of the h-* child
    • else if it's an e-* property element, re-use its { } structure with existing value: inside.
    • else if it's a u-* property element and the h-* child has a u-url, use the first such u-url
    • else use the parsed property value per p-,u-,dt-* parsing respectively

The difference between handling of p- properties and u- properties (there is no "and the h-* child has a p-name" clause) implies that p-name should be used unconditionally even if the child has none, yet the "else" at the end does does apply to p- properties.