For g≥2, n≥1, and d_1,…,d_n≥1 satisfying |mathbf d|=3g-3+n, let C(mathbf d) denote the primitive normalized Witten intersection number, and write C(3g-2) and C(2^3g-3) for the corresponding one-part and repeated-argument values. Nesting conjecture. C(3g-2)≤ C(mathbf d)≤ C(2^3g-3). The claim was motivated by numerical data for fixed genus, but the paper gives a counterexample: C(4,4)<C(3,5) contradicts the proposed ordering. Thus the conjecture is refuted.

Nesting conjecture for primitive normalized Witten intersection numbers

Guo, Yang, and Zagier formulate the nesting conjecture for admissible positive degree data, motivated by computations through genus $13$ . The supplied paper does not establish that the nesting bounds are false.

Known results

The lower bound $C(\mathbf d)\geq C(3g-2)$ is proved for all admissible data; it was previously known for $n=2$ .
Uniformly in positive degree data, $C(\mathbf d)=1/\pi+O(1/g(\mathbf d))$ as genus grows.
The upper bound $C(\mathbf d)\leq C(2^{3g-3})$ is not reported as proved.

March 2026 preprint: related monotonicity counterexample

The paper gives $C(4,4)<C(3,5)$ , disproving a lexicographic monotonicity property, and another counterexample to weak cross-length monotonicity. These do not contradict the nesting endpoints; no direct counterexample to the nesting conjecture is reported.

Current status (as of September 2026): the lower nesting bound is claimed proved, while the upper bound and any refutation of the nesting conjecture remain open.

Nesting conjecture for primitive normalized Witten intersection numbers

Primary source

Progress summary

Solutions 0