The problem with answering your question, as has already been identified to a degree, is that it assumes there is a clear agreed definition of progressive rock, when there isn't. For example, I don't believe genuine progressive rock was made after 1973/4. Music made after that in a '70s prog' style is exactly that, and certainly not progressive rock as originally envisaged or as it should be defined. It is retro or conservative, the opposite of progressive.

So in reality, your question is 'is Dool's House the first prog LP by my definition of prog rock?'. Only you can answer that. My opinion is that Doll's House is post-psychedelic experimental rock and therefore is definitely progressive. I also think it is rock and not pop. Just because it displays some typical 1968 motifs doesn't mean it isn't progressive. It wasn't ever going to sound like 1970. It is similar to Second Hand's Reality from the same year, another LP that is really post-psych but yet to display the cliches of what became prog rock and therefore not generally considered a prog LP.

Another early contender is Nice's Thoughts Of Emerlist Davjack, actually very much a psychedelic LP, that just predicted some of prog's classical pretentions early and is therefore often considered.

Perhaps you should give us your definition of progressive rock?