Some background info on the "island of stability" is that it's based in the Nuclear Shell Model, which asserts that nuclei are built up of shells, which are composed essentially of quantum energy levels separated by energy gaps.
My understanding of the "island of stability" is that stability is achieved by having energy levels in a shell that are completely filled with protons and neutrons (assuming spherical nuclei). These are known as magic numbers (so scientifically named...). All of these things are made in particle accelerators, and their half-lives are tiny, but the thought is that 126 is a magic number so each heavier element leading up to that is a step towards that number, at which point we'll see if that actually holds true or not.
The other problem is that nobody can agree on how long long these half-lives actually would be. Probably because of how complex and theoretical this is all. A good interview on the subject can be found here. While I have an interest in this, and a background in Chemical Engineering I don't fully-understand this.
Oh one more thing worth noting that's talked about in this article: There's all sorts of decay and reaction by-products going on when you're trying to synthesize an element. Without a good separation mechanism you might not easily find what you're looking for and end up with a low signal-to-noise ratio. It seems like they upgraded some equipment, which means in the future separation various isotopes and positively identifying them will be easier. Which is also very exciting to me.