According to a report from the Korean ChosunBiz LG had plans to adopt a flexible edge-type display in its next flagship phone, the G6. But LG Display could not produce enough such displays as it does not have enough capacity - and the Korean company will adopt a flat LCD display.
LG Display has been developing flexible OLED display technologies for a long time, and back in January 2015 the company already demonstrated the prototype you see above - a 6" plastic OLED display that curves on both edges.
LG currently produces plastic-based OLEDs in its Gen-4.5 fab, with a monthly capacity of 14,000 substrates (more than double its capacity in the beginning of 2014). LG Display's flexible AMOLEDs were used in LG's mobile phones (the G Flex 2) and are currently used in wearable devices (such as the Watch Urbane) and in Apple's Watch and Watch Series 2. So LG's flexible OLED capacity is quite limited at this stage.
LG announced plans to expand its small OLED production - the E5 6-Gen (1500x1850 mm) flexible OLED fab was announced in July 2015, and this $900 million line will have a capacity of 7,500 monthly substrates - or 1.5 million 5.5" panels. The new fab is scheduled to begin mass production in the first half of 2017, though so we should not really expect devices using displays produced at the E5 line before 2018 - which means that LG Electronics will not be able to use these displays in a 2017 phone...