MM / Upper MM Best Placement into HBS

As someone looking to make the transition to PE from a BB (GS/MS/JPM), what PE funds typically have the best placement into HBS?

I would almost imagine the top MM places (Kelso, Crestview, THL, New Mountain etc.) have better placements than many of the MFs due to the partners potentially going to bat for you and writing recommendations. And how do the upper MM typically place relative? My guess is that it's really on a case by case basis but would love to hear about (CD&R, H&F, BC Partners etc.).

If anyone has any insight into hours / culture at many of these that's bonus points.