Internal-solution convergence conjecture for the data-driven Lanczos ROM

Let $\Sigma\subset\mathbb{C}$ be the set of spectral points, and let $V_0$ and $Q_0$ be the solution basis and Lanczos matrix corresponding to the background $q_0=0$. Let $T$ be the Lanczos tridiagonal matrix, $M$ the mass matrix, $b$ the input vector, $e_1$ the first coordinate vector, and let $u$ denote the true solution while $\mathbf{u}$ denotes the data-generated internal solution. Internal-solution convergence conjecture. For all $\lambda\in\Sigma$,

$$u=\lim_{m\to\infty}\mathbf{u}=\sqrt{b^*M^{-1}b}\,V_0Q_0(T+\lambda I)^{-1}e_1.$$

Equivalently, for any fixed $\lambda\in\Sigma$, as the number of data points in $\Sigma$ tends to infinity, the data-generated internal solution $\mathbf{u}$ converges to the true solution $u$. This conjecture asserts that background solutions and data-derived Lanczos coordinates recover the internal solution in the infinite-data limit; the supplied text does not state a proof or resolution.