No term a sum of consecutive earlier terms: must $\limsup a_n/n=\infty$? (Erdős #839)

Let $1\leq a_1<a_2<\cdots$ be an infinite strictly increasing sequence of positive integers with the property that no term $a_i$ equals the sum $a_j+a_{j+1}+\cdots+a_{j'}$ of one or more consecutive earlier terms (with $j\leq j'<i$). Is it true that $$\limsup_{n\to\infty}\frac{a_n}{n}=\infty?$$ Or even the stronger assertion $$\lim_{x\to\infty}\frac{1}{\log x}\sum_{a_n<x}\frac{1}{a_n}=0?$$

Acceptance. FULLY RESOLVES: a complete proof that every sequence with the no-consecutive-sum property satisfies $\limsup a_n/n=\infty$ (or the stronger vanishing-reciprocal-density statement); OR a counterexample — an explicit construction of such a sequence with $\limsup a_n/n<\infty$ (equivalently $a_n\ll n$), accompanied by a machine-verifiable certificate that no term is the sum of consecutive earlier terms. ADVANCES: improve Freud's upper-density record $19/36$ — construct a sequence with the no-consecutive-sum property of upper density strictly greater than $19/36$ (state $19/36$ as the bar), with a machine-verifiable proof that the construction has the property and attains the claimed density; or prove a non-trivial upper bound on the maximal possible upper density of such a sequence; or prove the $\limsup$ statement conditional on a clearly-stated hypothesis. Deliver the construction plus its verification, or the proof.