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Claim. Let $C_2$ be the connective constant of the square lattice $\mathbb{Z}^2$, the case $k=2$ of Problem 528. A. R. Conway and A. J. Guttmann, Lower bound on the connective constant for square lattice self-avoiding walks, prove

$$C_2\ge2.62,$$

the bound as the paper's abstract states it. The method is Kesten's bridge method: the authors enumerate the irreducible bridges of the square lattice exactly up to 40 steps and bound the number of bridges of fewer than 125 steps from below, and the generating-function inequality for bridges then gives the lower bound on $C_2$. Together with Alm's upper bound this places $C_2$ in $[2.62,2.696]$; the numerical value $C_2=2.6381585303279\ldots$ of Jacobsen, Scullard and Guttmann [JSG16], cited on the problem page, is an estimate and not a proof.

Covers. The lower bound $C_2\ge2.62$. Not covered: any upper bound, any $k\ge3$, and the value of $C_2$, which is not determined.

Depends on. Nothing in this wiki; the claim rests on the cited paper.

Acceptance. Refereed: A. R. Conway and A. J. Guttmann, Lower bound on the connective constant for square lattice self-avoiding walks, J. Phys. A 26 (1993), no. 15, 3719--3724. The site's commentary records the bound, but the site labels the problem OPEN, so that remark is not acceptance of the problem and the page lists no reviewed evidence. The proof is not compiled in this corpus.

Dating. The page is dated by the issue date in the publisher's record, 7 August 1993.