I've looked at the screenshots and may have the answer. I first thought that another starbase had already claimed the relic, but the tool tip in the first screenshot would show that.
I then looked carefully at the ring around the starbases and noticed something. For the one that works, the ring is well past the relic. For the one that doesn't work, the ring doesn't reach past one small part of the tile with the relic. The tile isn't 100% inside the ring, so I'm going to guess that means it is considered to be the same as not being in the ring. If it's not in the ring, it can't be studied.
That's the only answer I can think of.