Gaussian upper-bound conjecture for walk counts on a high-dimensional torus

Let $c_n^{\mathbb{T}}$ be the $n$-step weakly self-avoiding-walk count on the torus, let $\mu$ be the growth constant on $\mathbb{Z}^d$, and let $V$ be the number of vertices in the torus. Gaussian upper-bound conjecture. Let $d>4$ and let $\beta\in(0,1]$. There exist $K,\alpha>0$, depending on $\beta$, such that, for all $n\geq 0$,

$$c_n^{\mathbb{T}}\leq K\mu^n e^{-\alpha n^2/V}.$$

The conjecture is intended to describe the onset of finite-volume effects at lengths of order $V^{1/2}$ and is presented as an upper bound paralleling the complete-graph behaviour. The supplied text gives no resolution.