The critical-exponent asymptotic conjecture for self-avoiding walks

Let $d\ge 2$, and let $G$ be a $d$-dimensional lattice. Write $\sigma_n$ for the number of $n$-step self-avoiding walks from a fixed vertex and $\kappa(G)$ for the connective constant. A critical exponent is a number $\gamma=\gamma_d$ for which there is a constant $A>0$ such that

$$\sigma_n\sim A n^{\gamma-1}\kappa(G)^n\qquad\text{as }n\to\infty.$$

The critical-exponent asymptotic conjecture. For every $d\ge2$, such a critical exponent exists, and

$$\gamma=\begin{cases}\frac{43}{32}&\text{when }d=2,\\1&\text{when }d\ge4.\end{cases}$$

The source notes that a logarithmic correction is expected when $d=4$. The stated values and the asymptotic formula are major predictions for self-avoiding walks; no resolution is supplied here, and the case $d=3$ is not specified by the displayed claim.