The 5770 already supports Metal. As such, it should also support Metal 2. Right now, 2012 Macs are the oldest listed as compatible with Mojave.
I did a search for 2010 Mac Pro and Mojave. 🙂 Here's are a few results, all mentioning the main specific line:
Mac Pro (late 2013 or newer, or mid 2010 and mid 2012 models with Metal capable GPU)
Here are the Macs that will work with MacOS Mojave - CNET
macOS Mojave will drop support for some older Macs released before 2012
Obviously, none of these are official Apple sites or statement, but they all seem indicate as long as the video card in Macs a bit older than the 2012 cutoff supports Metal, it should work. That means the AMD 5770 in my 2010 Mac Pro should allow Mojave to run.
From another site, this statement supposedly from the recent developer's conference:
Apple notes, however, that support for 2010 and 2012 Mac Pro models won’t be available right off the bat – instead that support will come “in an upcoming beta.”
So, we can't test to verify Mojave will run on a 5,1 Mac Pro until the public beta. And as mentioned by Grant, this is still preliminary and could change.