The true DGP is \begin{equation} y=\alpha_0 + \alpha_1 x_1 + \dots + \alpha_k x_k +\epsilon, \quad \epsilon\sim \mathcal{N}(0,1)\label{eq:1} \end{equation} but we instead estimate
\begin{equation} y=\alpha_0 + \alpha_1 x_1 + \dots + \alpha_k x_k + \dots + \alpha_{k'} x_{k'} \epsilon, \quad \epsilon\sim \mathcal{N}(0,1)\label{eq:2} \end{equation} where $k'>k$. The observations are fixed design.
Is it true that the standard error of the out of sample prediction of the estimated function (with spurious regressors) at some fixed $(x_1^*, \dots, x_{k'}^*)$ is (weakly) greater than the standard error of the out of sample prediction at $(x_1^*, \dots, x_k^*)$ using the correctly specified model? If so, how can I prove this?