Cycles hair transparency issue (regression due to Embree)

When you assign a transparent material to the hair strands in Cycles/Embree, black border issues appear. It gets worse when facing the hair strands.

This is a cube with hair emitted from its faces. I set the hair material to transparent. The radius from the tip tapers to smaller values so this is like rendering cones in this example.

You should only see the cube, not the weird black lines (and that 3 formed by two cones seen from the top)

This issue did not exist (at least not to that extent) before the integration of Embree in Cycles.

@brecht do you have any idea what could be the issue here? And is that rather on Cycles or Embree side? If you think it’s on Cycles side, any pointer to the class where you think I could have look is welcome.

I opened a bug report here #86500 - Cycles: tapering hair transparency fails in '3D Curves' mode [regression] (probably caused by embree) - blender - Blender Projects with example files.