Let $\rho>0$ be a smooth realvalued function such that $\rho=1$ outside the unit interval $(-1,1)$. For each $t>0$, let us denote by $\{\lambda_n(t)\}_{n=1}^{\infty}$ and $\{\phi_n(t;x)\}_{n=1}^{\infty}$ the eigenvalues (written in strictly increasing order) and eigenfunctions associated to the equation $$ -\partial^2_x \phi_n(t;x) = \lambda_n(t) \rho(x) \phi_n(t;x) \quad \text{on $(-t,t)$},$$ subject to Dirichlet boundary condition $\phi_n(t;\pm t)=0$.

Suppose that $f\in C^{\infty}_0((-1,1))$ and that
$$ \int_{-1}^{1} f(x) \phi_n(t;x) \,dx =0 $$for **some** $n \in \mathbb N$ and **all** $t>1$. Does it follow that $f=0$?