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Claim. Let $1=d_1<\cdots<d_{\tau(n)}=n$ be the divisors of $n$. Terence Tao's note "On the sum of reciprocals of gaps between divisors" (dated 9 September 2025) proves that, assuming the qualitative Hardy--Littlewood prime tuples conjecture (every admissible tuple $h_1,\ldots,h_k$ has infinitely many $n$ with $n+h_1,\ldots,n+h_k$ all prime), the ratio

$$\frac{\sum_{1\le i<j\le\tau(n)}\frac{1}{d_j-d_i}}{1+\sum_{1\le i<\tau(n)}\frac{1}{d_{i+1}-d_i}}$$

is unbounded as $n$ ranges over the natural numbers, so the inequality the problem asks about holds with no absolute constant. The construction takes $n$ to be the product of the primes $t-h_1,\ldots,t-h_K$ for an admissible tuple of diameter of order $K^2$ with consecutive elements at least of order $K$ apart, and $t$ chosen so that all $K$ shifts are prime, which is where the hypothesis enters. The pairwise sum over these $K$ prime divisors is then of order $\log K$ times the consecutive sum (the one-dimensional Coulomb-energy lower bound against roughly equal spacing), and a count of near-collisions among products of the primes keeps the other divisors from spoiling the ratio. The note remarks that the statement's upper summation limit should be $j\le\tau(n)$ rather than $j<\tau(n)$ as printed in [Er98], which does not affect the question, and that the numerology nearly allows the hypothesis to be removed.

Hypothesis. The qualitative prime tuples conjecture above is unproved; the claim is therefore conditional and derives no standing by itself.

Standing. The site's curator, Thomas Bloom, records in the problem's remarks that Tao disproved the statement assuming the prime tuples conjecture; the site's label rests on Larsen's unconditional disproof, so the remark acknowledges this result without settling the problem through it. The note is not refereed and no Lean proof of it is recorded, so the claim is claimed. The construction was made unconditional by Larsen's disproof, which repeats it over many scales.