AS of Skype 22.214.171.124, this solution does seem to work. I just tried it. Even though there is no checkbox to "Remember Me", "Remember Password", "Keep me signed-in", etc. upon logging-in, the default behavior seems to be to keep the user signed-in. Unfortunately, I tried one of the other solutions recommended here first (launching Skype as Administrator), so I'm not sure which action fixed it. I'll report back if I'm able to make a reliable determination at any point in the future. Thanks for this!