Let $\Phi$ be a subharmonic function on $B_{2}^{n}\times T^{k}=B_{2}\times\mathbb{R}^{k}/\epsilon\mathbb{Z}^{k}$ equipped with the Euclidean metric $g=\sum_{1}^{n}dx_{i}^{2}+\sum_{1}^{k}dy_{j}^{2}$, where $0<\epsilon\ll 1$. Let $v$ be the averaging function of $\Phi$ over the $T^{k}$ fibres. Assume $\mathchoice{{\vbox{\hbox{$\textstyle-$ }}\kern-7.98003pt}}{{\vbox{\hbox{$\scriptstyle-$ }}\kern-6.26338pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.6363pt}}{{\vbox{\hbox{$\scriptscriptstyle-$ }}\kern-5.45924pt}}\!\int|\Phi|\lesssim 1$ and a Lipschitz bound $\text{Lip}(v)\lesssim 1$, then on $B_{1}\times T^{k}$ we have $\Phi\leq v+C\epsilon^{1/2}$.