Is the $I3322$ quantum maximum reached in finite dimension?
This text mentions work from 2024 on. Those parts were partly written without a live literature search and have not been checked by an editor yet.
In plain words
$I3322$ is the simplest Bell test beyond the standard one, with three measurement choices per side. Numerics suggested that reaching its quantum maximum needs systems with infinitely many levels; two September 2026 preprints claim proofs that await independent checking.
Precise statement
For the Collins-Gisin $I3322$ Bell functional (two parties, three two-outcome measurements each), determine whether the supremum over tensor-product quantum strategies, approximately 0.25087538 (Pal and Vertesi, PRA 2010), is attained by any finite-dimensional strategy, and how the local dimension needed to come within $\epsilon$ of it grows. The residual task is verification of the 2026 claims: non-attainment in finite dimension, attainment by an infinite-dimensional strategy on $\ell^{2}(Z) x \ell^{2}(Z)$, and required dimension of order $\log(1/\epsilon)$.
What would settle it
Refereed or formally machine-checked confirmation of the non-attainment proofs, including their analytic parts.
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
Two independent preprints of August-September 2026 claim non-attainment in finite dimension (Douglas, arXiv:2609.05555; Krohn-Grimberghe, arXiv:2609.29693, computer-assisted, value bracketed to width $10^{-55}$); neither is refereed.
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