We are all equals, just at different points in our timelines.

I myself took much time to fully accept and embrace special relativity.

Ok, so let's say we use a distant pulsar which is conveniently on the axis of the disks plane. We have three views to consider.
1) What does the pulsar see from the disk?
2) What does the central disk observer see?
3) what does the edge observer see?

We start with 1)
The pulsar emits a pulse which we know travels through the same amount of space to get to both lights on the disk. They do receive the signal at the same time and therefore they emit a flash at the same time. If we tell the light operators to turn on the light at one pulse and then turn it off at the next pulse, the lights will be synchronized as far as the pulsar is concerned. If the light operators are told to leave the lights on for 1 full second, the pulsar will see the edge light as being lit longer, but it lights up at the same time.

Now we move to 2)
The central observer receives a pulse and turns on his light. He sees his light immediately, but the light from the edge will have a slight delay from the travel time it has to cover. Again, if we are leaving the light on for x amount of time on each pulse, he too will say the edge light is being left on too long.

Lastly, observer 3)
This person on the disk edge will also say the central light is slightly delayed, due to travel time. He will also say that the central operator is turning his light off early if we are using a fixed 'on' time.

So here we can see that the lights can remain synchronized for a far away observer all while the clocks are running at different speeds. We achieve this by telling everyone to switch the light on when one pulse is detected and turn the light off on the next pulse. This also implies that the observer on the edge will read a pulse that is shorter than the other observers, but this is of little importance as we assume the pulses are short enough to serve as instant markers.

p.s. I didn't proof read this, so some scrutiny is invited.