Minimal non-zero signed reciprocal sum Σ δ_k/k with δ_k ∈ {−1,0,1}: how small can it be? (Erdős #317)

Two questions about signed sums of the reciprocals $1,\tfrac12,\ldots,\tfrac1n$ with coefficients $\delta_k\in\{-1,0,1\}$. (1) Is there a constant $c>0$ such that for every $n\ge 1$ there exist $\delta_k\in\{-1,0,1\}$ ($1\le k\le n$) with $$0<\left\lvert\sum_{1\le k\le n}\frac{\delta_k}{k}\right\rvert<\frac{c}{2^n}?$$ (2) Is it true that for all sufficiently large $n$, every choice of $\delta_k\in\{-1,0,1\}$ satisfies $$\left\lvert\sum_{1\le k\le n}\frac{\delta_k}{k}\right\rvert>\frac{1}{[1,\ldots,n]}$$ whenever the left-hand side is non-zero, where $[1,\ldots,n]$ denotes the least common multiple of $1,\ldots,n$?

Acceptance. FULLY RESOLVES: complete proofs (machine-checkable preferred, else full written) settling BOTH parts — for (1), either exhibiting the constant $c>0$ with a proof that a non-zero signed sum below $c/2^n$ always exists, or proving no such $c$ exists (so the minimal non-zero value is superpolynomially larger than $2^{-n}$, as van Doorn's heuristic suggests); and for (2), a proof of the strict inequality for all large $n$, or an explicit infinite family of $n$ and sign patterns with a non-zero sum equal to $1/[1,\ldots,n]$. ADVANCES (each independently checkable): (a) improve the Kovač–van Doorn upper bound for (1) strictly below $2^{-n(\log\log\log n)^{1+o(1)}/\log n}$ with proof; (b) prove a matching lower bound on the minimal non-zero value, ruling the $c/2^n$ form in or out; (c) resolve (2) in either direction for a specified range or family of $n$; (d) extend an exact table of the minimal non-zero $\lvert\sum_k\delta_k/k\rvert$ to a new record $n$ with the search code and an optimality certificate. Deliver the proofs, the improved bound with proof, or the code + certified extremal values.