It seems Android developers are toying with the ability to cast a phone's display while the screen is off, based on a recent commit found in the Android Open Source Project (AOSP). Although it is worth mentioning that the Sony engineer-provided commit is one that has yet to be merged and therefore it is not something that is guaranteed to make its way to a future version of Android. For now, it just remains a possibility that is in testing.
For reference, generally speaking there are two main ways in which an Android phone owner can cast their display. The first, is the actual Cast feature that is built-in with most apps. This option is highly-integrated at the app level and allows users to fling content to a secondary screen and continue using their phone in the usual manner. Including the option to turn off the display while still maintaining to cast content. The second way is far more rudimentary in its approach as it involves the user hitting the "Cast" icon found in the Quick Settings menu (shown in the image above). Using this method results in the entirety of the display cast to a second screen. While this is often an ideal way to navigate a phone's interface while also 'mirroring' the display on another screen for others to see, it can also be of benefit when casting content through an app that does not otherwise support casting.
It is this second approach, and latter use case, that is of particular interest in this latest commit, as when casting the entire screen – even if it to play video – the screen on the phone needs to remain on for the mirrored feed to remain active. Essentially, this commit looks to remedy this issue by allowing the option to maintain the cast connection even if the display is turned off – in much the same way as how the app-based Cast solution currently works. The immediate and obvious benefits of such an implementation is the ability to conserve battery life, and also to permit the phone to continue operating with less heat generated.