Finite cotype problem for $\ell_{2}\widehat\otimes_{\pi}\ell_{2}\widehat\otimes_{\pi}\ell_{2}$

The problem asks whether the threefold projective tensor product of Hilbert sequence spaces has finite cotype. The public literature explicitly described this as open, and no verified resolution was found.

Known results

• Briët, Naor, and Regev (2012) proved failure of finite cotype for $\ell_{p_{1}}\widehat{\otimes}\ell_{p_{2}}\widehat{\otimes}\ell_{p_{3}}$ when $\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}\leq 1$; the present case has sum $\frac{3}{2}$.
• A 2016 paper on projective tensor products stated that the $\ell_{2}$ threefold case remained unknown.

Community submission (unverified), September 13, 2026

A submitted one-line claim says the space has no finite cotype. It supplies no proof or supporting reference, so this is not evidence of a resolution.

Current status (as of September 2026): The question remains unsettled; the classical parameter-range results do not cover $\ell_{2}\widehat{\otimes}_{\pi}\ell_{2}\widehat{\otimes}_{\pi}\ell_{2}$, and the September 13, 2026 community claim is unverified.