When quantizing rotations, rounding to nearest is not necessarily optimal.
Consider the following normalized quaternion:

0.000300531 (scalar)
0.995283
-0.0967456
-0.0072609

Quantizing to 8 bits with identity ball map (i.e. dropping the scalar, as done in spz) by rounding each of the remaining three components to nearest gives me the following representation:

254/127.5-1
115/127.5-1
127/127.5-1

Reconstructed normalized quat is (note the large difference on the scalar):

0.0774434 (scalar)
0.992157
-0.0980392
-0.00392151

Which gives me the following reconstruction errors:

• Geodesic quaternion distance (degrees): 8.85895
• Rotation angle distance (degrees): 8.84882
• Axis angle distance (degrees): 0.212142

However, the following quantized representation:

255/127.5-1
115/127.5-1
127/127.5-1

Gives me the following reconstructed normalized quat:

0 (scalar)
0.995221
-0.0975706
-0.00390277

Which has the following errors:

• Geodesic quaternion distance (degrees): 0.399584
• Rotation angle distance (degrees): 0.0344378
• Axis angle distance (degrees): 0.19881

The error is 22 times smaller, just by selecting another rounding direction.

I couldn't find a systematic way of finding the right rounding direction other than trying all 8 possibilities for rounding the three components.

I will submit a PR shortly with code that does that.