Problem of the coexistence of several non-Hermitian observables in $\mathcal{PT}$-symmetric quantum mechanics

During recent developments in quantum theory it has been clarified that observable quantities (such as energy and position) may be represented by operators $\mathrm{\Lambda }$ (with real spectra) that are manifestly non-Hermitian in a preselected friendly Hilbert space ${\mathcal{H}}^{\left(F\right)}$. The consistency of these models is known to require an upgrade of the inner product, i.e., mathematically speaking, a transition ${\mathcal{H}}^{\left(F\right)}\to {\mathcal{H}}^{\left(S\right)}$ to another, standard, Hilbert space. We prove that whenever we are given more than one candidate for an observable (i.e., say, two operators ${\mathrm{\Lambda }}_0$ and ${\mathrm{\Lambda }}_1$) in advance, such an upgrade need not exist in general.