Can a custom post type have a Parent Page?

I’ve seen a lot of posts out there in the Webiverse about assigning a page as a parent of a custom post type. After four hours I can’t find a solution and need some help. I’ve create an “About” page that is the parent of an “Our People” page. I’ve also create a custom post type called “People”. And I’ve created a custom page template for the “Our People” page. The Permalink structure looks fine until you get to the single “people” page. For Example: John Smith’s Page, the permalink isn’t correct.

Getting hierarchical custom post type permalinks to work just like pages

I’ve dug through every question here on custom post type permalinks, but most seem to be either problems with custom taxonomy rewrites, or the obvious missing of flush_rewrite_rules(). But in my case, I’m only using a custom post type (no taxonomy), set to be hierarchical (so I can assign parent-child relationships), with the proper “support” for the attributes metabox, etc, etc. I’ve flushed rewrite rules a thousand different ways. I’ve tried different permalink structures. But child URLs always result in 404!