I completely disagree with that statement. It's well known that with LiPo batteries a sustained full charge or deep discharge will reduce the lifespan of the battery. It is true that they don't have a memory effect. I think general consensus is to avoid discharging a pack below 20% and above ~80%ish to prolong pack life and get the most cycles from it.
Really good info here:
http://batteryuniversity.com/learn/article/how_to_prolong_lithium_based_batteries
Cliffs Notes:
Most Li-ions charge to 4.20V/cell, and every reduction in peak charge voltage of 0.10V/cell is said to double the cycle life. For example, a lithium-ion cell charged to 4.20V/cell typically delivers 300–500 cycles. If charged to only 4.10V/cell, the life can be prolonged to 600–1,000 cycles; 4.0V/cell should deliver 1,200–2,000 and 3.90V/cell should provide 2,400–4,000 cycles.
In terms of longevity, the optimal charge voltage is 3.92V/cell. Battery experts believe that this threshold eliminates all voltage-related stresses; going lower may not gain further benefits.
Similar to a mechanical device that wears out faster with heavy use, the depth of discharge (DoD) determines the cycle count of the battery. The smaller the discharge (low DoD), the longer the battery will last. If at all possible, avoid full discharges.
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Also, it almost sounds like your problem is voltage sag under high load causing the LVC(low voltage cutoff) to prematurely activate. LVC is built in to the BMS(battery management system) to protect the battery when it's approaching an empty state. I believe on the advanced LCD controller you can adjust what the cutoff voltage is.