Tim Penttila reminded me that in my previous post, I haven’t done $x=8$ (since $(q^2+1)/2 = 8.5$). So I’ve slightly altered my program, changing the size of my putative solution to 168, and ran it again. This time it took much longer to solve, hours rather than minutes, but it came back with “infeasible”. This means that there are no Cameron-Liebler line classes of $PG(3,4)$ with parameter 8.