One cookie is not working.

I am on a website that is highly trustworthy. That website's cookie is stored in my Safari Version 10.1.2 (12603.3.8). Nonetheless, when I go to a particular webpage at that website, I get this:

The red-letter text is what the webpage reports. And, as we see, I have told Safari to accept the cookie here.

One possibly interesting detail is that, the first two times, I clicked the button marked "Manage Website Data," Safari opened the cookie window and nothing appeared. I waited and waited, and Safari displayed no cookies whatsoever. After starting over at the website, I tried a third time, and this time "Manage Website Data" brought me to the window where all the cookies are supposed to be. This time the cookies were there. That is how I know the cookie for this website is in my Safari.

Why would Safari refuse to cooperate with a known, prestigious website?

TIA