Thanks for the message. Yes, the KDL55HX729 supports WiFi Direct. Make sure that the TV has latest software version installed. The latest software version available for your TV is PKG4.021AAA. You may refer to this link to download the latest software version. You may refer to this article for information on how to setup the TV to connect a Wi-Fi Direct capable device.
Based on the specifications of the TV, it does not support Miracast. You may check our eSupport website for information if your TV will support Miracast in the future.
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