Had baby at the GP with conjunctivitis months ago. Eye drops and oral antibiotics (swabs showed bacterial infection) and weeks went by and her R eye just was not clearing. Referred to eye clinic for paeds review, who diagnosed a blocked tear duct. Was told to do regular massage, keep it clean, and hope it cleared on it's own. I was to go back when she was 1 if not cleared yet, and they'd try to clear it surgically.
Fast forward a few months to a few weeks ago. I was demented with it. Constantly stuck shut, was sick to death of hearing "oh look, she's got a sore wee eye" 
as if I didn't know. Just over 2 weeks ago, we flew long haul for a holiday. I took a few pictures during the flight, and deleted them because her eye looked so bad. Got to our destination, woke up on the first day and THE EYE WAS CLEAR! I couldn't believe it. Unless it was a total coincidence, I can only assume the pressure on the flight cleared the blockage, which might explain why it was so bad on the flight.
Not looking for advice (because the eye has been perfect since), and not suggesting anyone book a flight, but just wanted to give hope to anyone dealing with a similar issue in their wee one. I reckon I've had about 3 months of cleaning a sticky funky eye, and I couldn't be more thrilled.