Alexa SmartThings skill won't discover Samsung TV

I have a Samsung TV, Model UN40J5200AF (which I know is “old”). I’m trying to get it linked to my Alexa device.

I have the Smarthings app on my phone, and I’ve successfully added the TV to that app. I’ve enabled the Smartthings skill in my Alexa app. I’ve linked the skill to my Samsung account. But the skill will not discover the TV.

I previously added my Samsung refrigerator to SmartThings, and I can see the fridge in my Alexa app, so I know I did something right then. I’ve ensured that the phone and the TV are both connected to the same wireless network in the house.

And I have only one Samsung account, so it can’t be that I’ve linked Alexa to the “wrong” Samsung account.

Can you think of anything I’m overlooking?