Yes it is a soft compound tyre designed to work on hard surfaces hence the name Carlsbad which was a very hard skatey surface apparently. I find tyres an interesting question, everyone has their own idea of what works. Someone once said that a 'mud tyre' works well at Nepean but that would be a hard compound wide spaced knob.
Personally, the best tyre I have used on my YZ125 is the Motoz X-Circuit. It feels incredibly stable under brakes, hooks up in loose corners and still works well when it's dry and hard. Way better than the Dunlop MX51 I was using before that. And much better in loose conditions than the Kenda. I reckon the Carlsbad would be good on an oiled dirt surface but I don't rate it in loose stuff...
But that's just me and I've not tried many others.