Limit points of normalized prime gaps: is $S=[0,\infty]$ for $(p_{n+1}-p_n)/\log n$? (Erdős #5)

Let $p_n$ denote the $n$th prime, and consider the normalized prime gaps $(p_{n+1}-p_n)/\log n$. Fix a real constant $C\ge 0$. Is there always an infinite sequence of indices $n_1<n_2<\cdots$ such that $$\lim_{i\to\infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C?$$ Equivalently, let $S\subseteq[0,\infty]$ be the set of all limit points of the sequence $\big((p_{n+1}-p_n)/\log n\big)_{n\ge 1}$. The problem asks whether $S=[0,\infty]$, i.e. whether every value $C\in[0,\infty]$ is attained as such a limit point. Since $S$ is closed, this is equivalent to asking whether $S$ is everywhere dense in $[0,\infty]$.

Acceptance. FULLY RESOLVES: a complete proof that $S=[0,\infty]$ — that for every $C\in[0,\infty]$ there is an infinite subsequence $n_i$ with $(p_{n_i+1}-p_{n_i})/\log n_i\to C$ — with a machine-checkable formalization (Lean/Coq) preferred, otherwise a full written proof; OR a disproof exhibiting a specific $C\ge 0$ that provably is not a limit point. ADVANCES (each requires a complete proof and must strictly improve on the best result stated in the background — currently: at least one third of $[0,\infty)$ lies in $S$, with bounded gaps, due to Merikoski): prove that a subset of $[0,\infty)$ of Lebesgue density strictly greater than $1/3$ lies in $S$; or prove $[0,c]\subseteq S$ for an explicit $c$ exceeding the currently known range; or establish a new structural feature of $S$ (a specific interval strictly larger than currently known contained in $S$, or a smaller uniform bound on the gaps of $S$). Deliver a machine-checkable proof or a complete written proof with explicit constants.