Older versions of the HTM whitepaper had a step in TM which allowed connecting with earlier timesteps (I don’t recall the exact implementation off hand, I’ll looked it up from my notes later if you are interested).
Personally, I think this is a function of a temporal pooling layer, as you mentioned. If you think of a sequence as an object, apical feedback of that object represented in a pooling layer cause all the elements of the sequence (or section of the sequence, depending on the pooling implementation) to become predictive.