Let $\Sigma$ be a compact smooth surface with boundary. Is it true that the supremum

$$\sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ smooth Riemannian metric on $\Sigma$} \}$$

is finite? Here, $\lambda_1(\Sigma,g)$ denotes the first eigenvalue of the Laplacian associated to the metric $g$ with Dirichlet ($=0$) boundary condition. This is true if $\Sigma$ has no boundary, as shown by Yang-Yau, for instance.

According to the answer by Gabe (see below), this is false for a cylinder. Is it false for every compact surface with boundary?