How universal are asymptotics of disconnection times in discrete cylinders?

We investigate the disconnection time of a simple random walk in a discrete cylinder with a large finite connected base. In a recent article of A. Dembo and the author it was found that for large $N$ the disconnection time of $G_N\times \mathbb{Z}$ has rough order $\mathrm{\mid }{G}_N{\mathrm{\mid }}^2$, when $G_N=(\mathbb{Z}\mathrm{/}N\mathbb{Z}{)}^d$. In agreement with a conjecture by I. Benjamini, we show here that this behavior has broad generality when the bases of the discrete cylinders are large connected graphs of uniformly bounded degree.