The paper says that if a strongly sub-quadratic solution exists than the exponential time hypothesis (that SAT cannot be solved in subexponential time) is invalidated.
That's very interesting, but it's not a proof that no strongly sub-quadratic solution exists for SED.
Note that exponential time hypothesis is strictly stronger than P!=NP. Even if SAT can't be solved in poly time, that doesn't mean it can't be solved in subexponential time. There are functions that lie between polynomial and exponential...
Of course, the paper was careful to explain it, but the media summary...
Edit: I was interested to learn about the notion of strongly quadratic; there are O(n^2/log n) solutions to SED, but this paper is casting doubt on solutions with time complexity O(n^(2-delta)) for any delta > 0. Another commenter mentioned a method to solve SED like this.