Weakly self-avoiding walk free-energy correction

For $d\geq 3$, let $f^{\mathrm{wsaw}}(u)$ be the weakly self-avoiding walk free energy and let $\lambda_d$ be the constant defined from the simple random walk Green function. Weakly self-avoiding walk conjecture. There are constants $a_d>0$ such that, as $u\downarrow 0$,

$$\lambda_du-f^{\mathrm{wsaw}}(u)\sim\begin{cases}a_3u^{3/2},&d=3,\a_4u^2\log(1/u),&d=4,\a_du^2,&d\geq 5. \end{cases}$$

This predicts the first correction beyond the linear weak-coupling term in dimensions three and higher. The source presents it as a conjectural higher-order expansion and does not provide a proof.