Bases as solutions to HODEs

Let $(\mathcal{F}_{\mathbb{M}}^{\mathbb{M}},{\Greekmath 011A})$ be a Hilbert function space. Assume that $\mathcal{E}=\{e_i\}_{i\in I}\subset \mathcal{F}_{\mathbb{M}}^{\mathbb{M}}$ is an orthonormal basis of $\mathcal{F}_{\mathbb{M}}^{\mathbb{M}}$, with

$$\overline{\mathcal{E}}=\mathcal{F}_{\mathbb{M}}^{\mathbb{M}},$$

where the index set $I$ is countable or uncountable. Bases as solutions to HODEs. There exists a unique linear HODE, potentially of infinite order, whose fundamental set of solutions obtained via Wronski orthogonalization is $\mathcal{E}$. The conjecture proposes a general converse to the construction of orthonormal bases from HODE solution sets, extending the observed finite-dimensional or ODE-based phenomenon to arbitrary Hilbert function spaces and index sets; no resolution is provided here.