I did delete the app and reinstalled, with no change in behavior.
My phone is iOS 9.3.3 (iPhone 6S Plus)
My computer is Windows 10, Scrivener 1.9.5.0
When I created a new project on my phone, everything seems to work fine. When I create a new project in Windows, it still seems to work fine. The problem seems to be with pre-existing projects I have, which were created under the previous version of Scrivener for Windows.
I’m still not able to get it syncing in any reliable fashion. I created a new project in Windows, saved into the Dropbox folder on my computer. I added text into one of the files. Then I waited for Dropbox to finish syncing. On my phone, I linked the app to Dropbox and then synced. The project came over, but when I open that file, it’s empty. If I go back to Windows and open the project again, the file still has text in it.
Now I can repeat the above process, and sometimes it works fine. Most of the time, it doesn’t. If I have multiple files in the project, some of them will always work, others will work intermittently.