Except...... it does.....
Ekko's next basic attack has Range icon 300 bonus range, deals bonus magic damage, and causes him to blink to his target. The bonus magic damage is classified as ability damage.
That's literally what E is; A Basic attack stimulant. In other words, E is a Buff, not an active skill. It's a buff to an auto attack. Lich bane procs on the auto attack, as does W's
Ekko's basic attacks deal 3% (+ 3% per 100 AP) of the target's missing health bonus magic damage to targets below Health icon 30% of their maximum health. The damage has a minimum threshold of 15 and is capped at 150 against Minion icon minions and Monster icon monsters.
In other words, Ekko with ~ 300 AP:
Assuming just E:
E procs Lich Bane.
On Hit E damage: 270
On Hit Lich Bane Damage: 264.5 damage
If the target has 1100 HP, the W by itself won't proc.
However, if he hits Q or R, it's part of a certain death equation.
If the target is below 80% HP, it will execute with W.