A strong multiplicity one theorem for dimension data

We call the dimension data $\mathscr{D}_{H_{1}}$ and $\mathscr{D}_{H_{2}}$ of two closed subgroups $H_{1}$ and $H_{2}$ of a given compact Lie group $G$ {\it almost equal} if $\mathscr{D}_{H_{1}}(\rho)=\mathscr{D}_{H_{2}}(\rho)$ for all but finitely many irreducible complex linear representations $\rho$ of $G$ up to equivalence. When $G$ is connected, we show that: if $\mathscr{D}_{H_{1}}$ and $\mathscr{D}_{H_{2}}$ are almost equal, then they are equal. When $G$ is non-connected, $G^{0}\subset H_{1}\cap H_{2}$ is a trivial sufficient condition for $\mathscr{D}_{H_{1}}$ and $\mathscr{D}_{H_{2}}$ to be almost equal. In this case assume that $\mathscr{D}_{H_{1}}$ and $\mathscr{D}_{H_{2}}$ are almost equal but non-equal. We show strong relations between $H_{1}$ and $H_{2}$ and we construct an example which indicates that $G^{0}\subset H_{1}\cap H_{2}$ is not a necessary condition.