Windows Media Player will not play MP4 over HTTPS

Question

I have a website that serves video. These videos are transmitted via HTTPS as MP4 files. The problem is that I cannot get Windows Media Player to play it.

Videos are played in the tag <video>on the web page in all browsers. The video will play if I bind the URL to the VLC. Video plays if downloaded and then played in Windows Media Player. But if I point Windows Media Player to the URL (let's say someone clicked the link and Windows automatically opened the player), it just sits there with the message "Connecting ...".

I cannot find any magazines or other ways to understand why Windows Media Player does not like streaming video. Where should I look? What should I check?