Before I say anything else, that is a really nice rack.
I just built a chisel rack as well. Mine mounts on the wall behind the lathe (Jet 1642). I do have my cutting edge up. I put one back before I take another out. I do not carry a chisel to the lathe it is all right there. Right by that rack is a tray that holds my steb centers with drop in holes, pen mandrel, also a drop in. And all my other frequently used lathe accessories.
May I please ask a noob question? Could you please explain the "Death Trap"? Sorry if this is dumb.
Thank You
Andy