Is the least-prime-factor sum $\sum p(n)/n$ over every short window $\gg 1$? (Erdős #462)

Let $p(n)$ denote the least prime factor of $n$. It is known that there is a constant $c>0$ such that $$\sum_{\substack{n<x\\ n\text{ not prime}}}\frac{p(n)}{n}\sim c\,\frac{x^{1/2}}{(\log x)^2}.$$ Is it true that there exists a constant $C>0$ such that $$\sum_{x\le n\le x+Cx^{1/2}(\log x)^2}\frac{p(n)}{n}\gg 1$$ for all large $x$? In words: the mass of the summatory function grows like $x^{1/2}/(\log x)^2$, and the question asks whether that mass is spread out enough that every interval of length about $C\,x^{1/2}(\log x)^2$ near $x$ carries at least a bounded-below amount of it.

Acceptance. FULLY RESOLVES: a proof that such a constant $C$ exists (the windowed sum is $\gg 1$ for all large $x$), or a proof that no $C$ works (for every $C$, construct arbitrarily large $x$ whose length-$C x^{1/2}(\log x)^2$ window carries mass $o(1)$). ADVANCES: prove the windowed lower bound with the window length inflated by a slowly growing factor (window $C\,x^{1/2}(\log x)^2 g(x)$ with $g(x)\to\infty$), or establish the bound for a density-one set of $x$; or, computationally, evaluate the windowed sums over a large stated range of $x$ at a specified $C$ and report the minimum observed value together with the $x$ achieving it, as reproducible evidence. Deliver the proof or the evaluation code plus the tabulated windowed minima.