The reason why you have recovery issues at the end of LP is because the LP at that point is not stressful enough to disrupt homeostasis and thus drive strength gains like it did initially. As the beloved 3x5 sets are becoming less stressful and causing a decrease in strength gains, the intensity starts climbing (because you are not responding to the stress as robustly anymore). Eventually, because the stress of 3x5 is increasing insufficient, you get to some brutal, grindy, terribly reps, and you might reset once or twice, maybe add a light day, and shortly after switch (hopefully) to more appropriate intermediate programming.
Recovery is an issue at the end of LP not because you need more recovery, it's because you need more stress from your programming.