Critical-window susceptibility conjecture for weakly self-avoiding walk on a torus

Let $\chi^{\mathbb{T}}(z)$ be the torus susceptibility, let $z_c$ be the critical point, and let $V$ be the number of vertices in the torus. Critical-window susceptibility conjecture. Let $d>4$ and let $\beta>0$ be sufficiently small. For any $s\in\mathbb{R}$, there is a constant $K_s$, depending on $\beta$, such that

$$\chi^{\mathbb{T}}(z_c(1+sV^{-1/2}))\leq K_sV^{1/2}.$$

Together with the known lower bound, this would show that the susceptibility is of order $V^{1/2}$ throughout the critical scaling window, in parallel with the complete graph. The supplied text gives no resolution.