In my experience, specificity matters enormously when going heavy (5RM and especially lower). I simply must train heavier to be better at going heavier; I could never back-door into a heavier single even if my rep-outs (at 8+ reps) improved. Same was true–for me-- even for chinnups. As my BW chins improved my loaded chinnups did not.
The reverse is not true for me. That is, as my low rep ranges increase I’m usually able to rep out better (assuming the higher rep range is under 12 reps).
Thus, the greater my 1RM the better were my higher rep ranges (if below 12 reps).
My guess here is that the skill requirements and neural components are much bigger players at low rep ranges and thus need be trained for as per that specificity. And there my be a psychological component too at the heaviest loads.
Of course, all this may vary across a population of lifters, as per any bell curve. But many I know personally experience the same thing.
Recently, I have learned the incredible advantages of programming based almost wholly on Singles (for the main lifts) by using a Mark Chaillet approach to singles training (as described by Marty Gallagher) This has pushed my higher reps (when I test them) but I could care less about higher rep performance.