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This User opened a current Thread here at AMD Forum for the exact same problem as you are having.: Monitor going into power save mode during gaming
Note: But in that User's case he seems to have a underpowered PSU (Weak). You have a stong enough PSU (700 Watts) for your computer setup. Try using the configuration shown in that thread to configure your RX 580 using AMD Wattman. Please be aware it is advisable to uninstall any 3rd party software similar to MSI Afterburner since it can cause confict with AMD Wattman ( Even if Wattman is not activated via Radeon Settings).