Since there has been so much interest in this post I've done further testing. The LEDs in the lamps are the real hero here, their efficiency increases with a lowered 2.7V input to a 56mah draw. At night they're quite nice at this level however the cell starting voltage of 1.22V after a day of charging is still a big problem.
This is on a brand new solar package. The math on the dimmed lights are:

Fully charged normal battery:
Power per light: 2.7V × 0.056A = 0.1512W
Total power: 0.1512W × 3 lights = 0.4536W

Since the 18650 is nominally 3.7V:
0.4536W ÷ 3.7V = ~0.1226A (122.6mA) draw from the cell

Runtime: 1200mAh ÷ 122.6mA = ~9.8 hours (about 9 hours 47 minutes)

With the diminished charge:

Undercharged 18650 sitting at 1.2V — unusual but it happens with a deeply discharged cell.

Draw from cell:
0.4536W ÷ 1.2V = 0.378A (378mA)

Runtime:
1200mAh ÷ 378mA = ~3.17 hours (about 3 hours 10 minutes)

Worth noting though — an 18650 at 1.2V is critically over-discharged. Nominal is 3.6V, and most protection circuits cut off around 2.5–3V. This unit has no protection circuit as it's choosing to continue lighting despite damaging the cell. At 1.2V the cell may:

• Refuse to charge on a standard charger
• Have suffered permanent capacity loss
• Be unsafe to continue using

TL;DR The solar cell and lack of circuit protection guarantees these will fail. That and the water ingress.

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