Does the monitor that has the Web Player showing turn off, or is it random? Have you tried having the Web Player on your laptop screen and see if the same thing happen while the other monitors are turned on?
Further, what happens when a monitor shuts down? Does it lose signal, or shut down completely?
Please ensure that you have your browser up-to-date, and try with different browsers to see if the same thing happens. If it does, let us know your browser versions.
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same problem here with DELL Latitude 7490 Laptop connected with 2 EIZO monitors via Dell Business Dock WD15, used as 2nd and 3rd display, running in Chrome (Version 70.0.3538.77, 64 bit) on Windows 10.
As soon as the web player loads in the browser - i don't even have to hit play my second monitor turns off. I have all the latest browser and windows updates. Dell Latitude 7480 using a Dell Dock. The signal loss is as though the monitor has been has its HDMI cable removed, still has power, just not signal, then eventually goes into sleep mode. As soon as i close the browser tab, the monitor comes on again.