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Claim. Two lower bounds on the function $h_3(d)$ of Problem 934, from H. Kumar, B. Mohar and S. Pragada, An improved bound for the strong clique index of graphs, arXiv:2607.02698v1 (2 July 2026), 15 pages, cited as [KMP26] on the problem page. Lemma 3.1 (p. 9): the line graph of the odd graph $O_4=\mathrm{KG}(7,3)$, which is $4$-regular on $35$ vertices with $70$ edges, has diameter at most $3$, so $h_3(4)\ge71>54=4^3-4^2+4+2$, and the truncated Witt graph likewise gives $h_3(15)\ge3796>3167$ (p. 10); these refute the 2022 conjecture $h_3(d)\le d^3-d^2+d+2$ of Cambie, Cames van Batenburg, de Joannis de Verclos and Kang (Conjecture 1) at $d=4$ and $d=15$. Theorem 1.11 (p. 4): $\liminf_{d\to\infty}h_3(d)/d^3\ge\frac{253}{225}$, that is, $h_3(d)>(1+\varepsilon)d^3$ for every $0<\varepsilon<\frac{28}{225}$ and all large $d$, by an infinite family built from projective planes over the two counterexamples; this refutes Conjecture 1 for all large $d$ and the upper asymptotic conjecture $h_t(d)\le(1+o(1))d^t$ (Conjecture 4) at $t=3$. The preprint's Problem 1.12 asks whether $h_3(d)\le\frac{253}{225}d^3$ for all large $d$, and it records that Conjecture 4 is undecided for $t\ge4$. Its "AI statement" (p. 13) declares the use of AI tools during the ideation phase and that the text is not AI-generated.

Covers. The lower bounds $h_3(4)\ge71$, $h_3(15)\ge3796$ and $\liminf h_3(d)/d^3\ge\frac{253}{225}$, hence the refutation of the two 2022 conjectures at $t=3$. Not covered: the value of $h_3(d)$ at any $d\ge4$ (the matching upper bound $h_3(4)\le71$ is the separate claim on the BitterLemma page), the leading constant of $h_3(d)$, and every $t\ne3$.

Depends on. No page of this wiki; the arguments are the preprint's own.

Standing. Claimed. The preprint is unrefereed (arXiv v1 only, 2 July 2026), the site's commentary does not mention it (page last edited 28 October 2025; label OPEN), and no referee or named reviewer is recorded. A thread post of 17 August 2026 (the discussion link above) cites Lemma 3.1 and Theorem 1.11 as the refutation of the displayed conjecture, and a comment of 30 July 2026 on the Korsky claim's thread calls the Korsky construction a generalization of the preprint's Lemma 3.3; neither is a review. No independent review of Lemma 3.1 is recorded.