With mu_a the biased-coin product probability measure on the product group G defined in Section 3, Conjecture 6 states: Given a>0, there exist C(a)>0 such that for u≥ 2, psi_mu_a(u)≤ fracC(a)√log u. The measure used in Section 3 is the product measure whose one-coordinate factor is frac1+a2delta_1+frac1-a2delta_-1.

Talagrand’s convolution conjecture

With $\mu_a$ the biased-coin product probability measure on the product group $G$ defined in Section 3, Conjecture 6 states: Given $a>0$ , there exist $C(a)>0$ such that for $u\geq 2$ ,

\psi_{\mu_a}(u)\leq \frac{C(a)}{\sqrt{\log u}}.

The measure used in Section 3 is the product measure whose one-coordinate factor is $\frac{1+a}{2}\delta_1+\frac{1-a}{2}\delta_{-1}$ .

Progress summary

Solved

A 2026 preprint claims to settle Talagrand’s conjecture, but its proof has not yet been independently verified.

Talagrand’s 1989 conjecture asks whether the biased product measure satisfies the sharp decay bound $\psi_{\mu_a}(u)\leq C(a)/\sqrt{\log u}$ for $a>0$ and $u\geq 2$ .

Known results

Y. Chen obtained the bound with an extra $\log\log u$ factor: $\psi_{\mu_a}(u)\leq C(a)\log\log u/\sqrt{\log u}$ .
Chen’s 2025 preprint established the conjectured decay up to a dimension-free factor $(\log\log u)^{3/2}$ and resolved the weaker question $\lim_{u\to\infty}\psi(u)=0$ .

August 2026 claimed proof

A preprint claims the exact $1/\sqrt{\log u}$ estimate on the Boolean hypercube, removing the extra logarithmic factor. It says the proof was discovered with the assistance of Odin Automatic AI Research Agent and reorganized by the authors; no independent verification, gap report, withdrawal, or retraction is reported in the retrieved sources.

Current status (as of August 2026): The conjecture is claimed solved by a 2026 preprint, but the claim remains unverified; earlier results retained an extra logarithmic factor.