Backup the system's "Personal Dictionary" on Android 7 and import on different phone? I'm using the Google Keyboard (Gboard) on my Nexus 6p.  I have a bunch of words in my system's Personal Dictionary (viewed by going to Settings -> Languages &amp; input -> Personal Dictionary), and when I setup a new phone (a replacement Nexus 6p), the dictionary did not transfer. 

In the Gboard keyboard settings (Settings -> Languages &amp; input -> Virtual keyboard -> Gboard -> Dictionary), there is an option to "Sync learned words", and after syncing on both the old and new phone, the Personal Dictionary on the new phone is still blank.

I've also tried removing all GBoard updates and data, removing Google accounts, re-installing and re-syncing everything with not improvement.

With the phone currently not rooted, is there a way to backup this data and import it on my new phone?

Update (12/30): 


I spent a few hours on the phone with Google Support yesterday, trying some of the stuff above plus some other things, and the end conclusion was that there is indeed a bug with the syncing, so they are going to escalate the issue for developers to look at. 
Some online (on the Nexus forums) reported the dictionary copying over when doing a restore from a different device when initially setting up the phone. I actually did this, but did get some generic failure message during my restore, so maybe that's why my dictionary is missing. I'm going to try a factory reset and restore again to see if it helps. 
As I mentioned below, the 3rd party utility didn't work for me, at least on two separate tries. The exported .DAT file was essentially 0 bytes.
I also checked my Google Dashboard, and under Android for my original device, there is an entry for "Android Dictionary" under "Applications with backups". This gives me hope that the data is in the cloud, though I'm going to refrain from formatting my old phone as long as I can (Google eventually wants it back since it is an RMA).
It is quite odd that it isn't syncing across devices. I know that recommending an application may not always be the best solution but I did just try Personal Dictionary Backup on my Xperia XZ running 7.0 and it was able to backup and restore my personal dictionary. 

Do keep in mind that it backs up to a .dat file on your internal storage. The folders name that it backs up to is "UserDictionaries". This means that you would have to digitally transfer the .dat file in that folder to your new device in a folder with the same name as well as installing the app on your new device.