Set $G_\eta:=\{(x,y)\in \mathbb{R}^2|-\eta<x<\eta, 0<y<1\}$. If $u_\eta\geq 0$ are a sequence of subharmonic functions defined on $G_\eta$ such that $$ \int_{G_\eta}|u_\eta|^2dx\wedge dy\leq C\eta, $$ where $C$ is some constant. Then can we show that $u_\eta$ is uniformly bounded on the line $0\times [\frac{1}{3},\frac{2}{3}]\subset \mathbb{R}^2$?

Set $G_\eta:=\{(x,y)\in \mathbb{R}^2|-\eta<x<\eta, 0<y<1\}$. If $u_\eta\geq 0$ is a sequence of subharmonic functions defined on $G_\eta$ such that $$ \int_{G_\eta}|u_\eta|^2dx\wedge dy\leq C\eta, $$ where $C$ is some constant, then can we show that $u_\eta$ is uniformly bounded on the line $0\times [\frac{1}{3},\frac{2}{3}]\subset \mathbb{R}^2$?