Are there infinitely many amicable pairs, and is $A(x)>x^{1-o(1)}$? (Erdős #830)

Call $a,b\in\mathbb{N}$ an amicable pair if $\sigma(a)=\sigma(b)=a+b$, where $\sigma$ is the sum-of-divisors function (equivalently, each of $a,b$ equals the sum of the proper divisors of the other); the classical example is $(220,284)$. Are there infinitely many amicable pairs? Moreover, if $A(x)$ counts amicable pairs with $1\leq a\leq b\leq x$, is it true that $$A(x)>x^{1-o(1)}?$$

Acceptance. FULLY RESOLVES: a complete proof that there are infinitely many amicable pairs together with a proof of the quantitative bound $A(x)>x^{1-o(1)}$ (for every $\epsilon>0$, $A(x)>x^{1-\epsilon}$ for all large $x$); OR a disproof of one of these, e.g. a proof that only finitely many amicable pairs exist, or a proven upper bound $A(x)\leq x^{1-\delta}$ for a fixed $\delta>0$ that refutes $x^{1-o(1)}$. Machine-checkable proof preferred, otherwise a full written proof. ADVANCES: a proven non-trivial lower bound on $A(x)$ (any explicit unbounded lower bound would be a breakthrough, since none is known); or a strict improvement of the exponential upper-bound factor of Pomerance [Po15] stated in the background, with full proof; or an extension of the exhaustive count $A(x)$ to a new verified height $x$ with a reproducible enumeration certificate. Deliver the proof/bound, or the search program plus the attained height and the certified count.