It seems the specular is weaker than the reference. I looked into it, and the fix is that the view direction should be traced from the center of the area light instead of where you sample the light source. Im quite confused by how it is correct though.

spp 8-128.

Send the relection ray in the half vector direction cdftheta = alpha + 1 1 - cos^alpha theta. Technically I should reflect v by h and send a ray in that direction? But this looks ok anyway.