I just tested a gmini 5200 mAh power bank. It seems to shut off after 24 s.
12 mA is not enough to keep it on, but 17 mA is. In other words, a 400 Ohm load is not keeping it on, but 300 Ohm does the job. A 100 ms pulse of 17 mA every 20 second keeps the power on. The power won't go on just by connecting it to something that would draw more than 17 mA. Only the button on the PB can turn the 5 V on.
Next test is with an Anker Astro E1 power bank. This one behaves strangely. Even 25 mA is not enough to keep it on. While on, it stays on for 30 s. When off, connecting a 300 Ohm resistor will turn it on, but it won't stay on even if the 300 Ohm is left on. I can't figure out how big current is required for it to stay on. This PB is probably only for charging. I bet it stays on, if the initial current is over 1000 mA (it can deliver 2000 mA) and then the current fades with an even rate, as it would, when a phone actually is charged. So an uneven load caused by an Arduino and a few servos and a timed small 50 mA peak might not keep this PB awake. Which is strange, because I got this PB in a Raspberry Pi set. Perhaps the Raspberry Pi draws continuously enough to keep the Anker PB awake.
The conclusion is that a gmini GM-PB052L works fine with an Arduino, if you just give it an extra load of 300 Ohm (17 mA) for 100 ms every 20 s.