When I initially read the headline I was stunned

Same. Especially the "side project" part

I mean, there isn't anything known as "the prime number conjecture". Goldbach's conjecture is a prominent conjecture about primes, but it's not "the prime number conjecture".

Goldbach's conjecture is possibly the most famous unsolved problem in mathematics. When people refer to prime number conjecture, this is the most obvious one everyone's thoughts will go to.

I'd say the Riemann hypothesis is more famous. Goldbach's conjecture didn't make the Milennium Prize Problems list, and Perleman's refusal of the award for his proof of the Poincaré conjecture popularized that list quite widely.

I thought it was just what they called the Prime Number Theorem before it was proved.

At first this is what I thought was solved - still cool regardless though!

It's the Erdős primitive set conjecture https://www.maths.ox.ac.uk/node/36408

if it was the conjecture, it would be in the headline

Goldbach's Conjecture is not the Prime Number Conjecture , because there is no such thing. There are several prime number conjectures.