1. By an inequality due to Robin (see (2.2) in Lagarias's paper), $$\sigma(n)\leq e^\gamma n\log\log n+\frac{n}{\log\log n},\qquad n\geq 3.$$ By Lemma 3.1 in Lagarias's paper, we also know that $$\exp(H_n)\log(H_n)\geq e^\gamma n\log\log n,\qquad n\geq 3.$$ Combining these two estimates, we infer that $$\sigma(n)\leq\left(1+\frac{1}{(\log\log n)^2}\right)\exp(H_n)\log(H_n),\qquad n\geq 3.$$ The fraction on the right hand side tends to zero (effectively), confirming the observation by the GCHQ Problem Solving Group. Explicitly, given any $\varepsilon>0$, we have $$\sigma(n)\leq(1+\varepsilon)\exp(H_n)\log(H_n),\qquad n\geq\exp\exp(\varepsilon^{-1/2}).$$
2. The barrier to prove Lagarias's inequality is the same as the barrier to prove the Riemann Hypothesis, since these two statements are equivalent. It is a difficult problem, and perhaps it is even undecidable from the ZFC axioms (of course most of us believe it is decidable).