The article clearly states this, but the title doesn't, so N.B.: the proof is conditional on exponential time hypothesis, that SAT requires exponential time. Of course we don't know whether it does.
What the article doesn't state clearly is that this assumes the strong exponential time hypothesis: it assumes that SAT cannot be solved in time 1.9999^n -- in other words that it's impossible to do better than the brute-force algorithm, which has complexity 2^n (up to polynomial factors). That's a very strong assumption.