True Contact is only available for Android < 6 (Xperia X is now using Android 7) and I didn’t find a similar app. What I did find is that all fields synchronize when using the “CardDAV-Sync”-App unfortunately this app is not yet open source, so it’s not possible to see how they implemented a work-around.
Contact pictures not syncing
A few days ago I switched from CalDAV-Sync and CardDAV-Sync to DAVdroid. Earlier today I created a contact on the phone and added a picture. When the contact syncs to my ownCloud server, the picture disappears. This did not happen with CardDAV-Sync.
The other direction works fine. I can create a contact on ownCloud, add a picture, and it syncs to the phone. I can also create a contact in KDE’s KAddressBook, add a picture, and it syncs to the phone.
I see there are some similar issues here about degrading images. I’ve opened a new issue because, in my case, the image is deleted.
Android: 4.4.4 (on Samsung Galaxy Note 4)
Seems to be an OwnCloud problem. Please follow up at https://github.com/owncloud/contacts/issues/1011.