Uniform stable approximation conjecture for the parametric approximation sets

Let $\mathcal{A}_{P}$ be the finite-dimensional truncation of the parametric Hilbert space $\mathcal{A}$, let $\boldsymbol{\Psi}_{P,M}$ denote the first $M$ sampling functionals in the approximation set $\boldsymbol{\Psi}_{P}$, and let $\mathcal{T}_{\boldsymbol{\Psi}_{P,M}}$ be their approximation operator. Fix the tolerance $\eta$ appearing in the conjecture.

Uniform stable approximation conjecture. The sequence $\boldsymbol{\Psi}_{P}$ is a stable approximation for $\mathcal{A}_{P}$: there exist $s=s(P)\geq 0$ and $C=C(P)\geq 0$ such that, for every $v_{P}\in\mathcal{A}_{P}$, there are $M\in\mathbb{N}$ and $\boldsymbol{\mu}\in\mathbb{C}^{M}$ satisfying

$$\|v_{P}-\mathcal{T}_{\boldsymbol{\Psi}_{P,M}}\boldsymbol{\mu}\|_{\mathcal{A}}\leq\eta\|v_{P}\|_{\mathcal{A}},\qquad \|\boldsymbol{\mu}\|_{\ell^{2}}\leq C M^{s}\|v_{P}\|_{\mathcal{A}}.$$

Moreover, the smallest such exponent and constant are uniformly bounded in $P$:

$$s^{*}:=\sup_{P\in\mathbb{N}}\{s(P)\}<\infty,\qquad C^{*}:=\sup_{P\in\mathbb{N}}\{C(P)\}<\infty.$$

This is the paper's main conjectural stable approximation result. Numerical experiments indicate excellent approximation and stability for the associated sets, whereas a rigorous analysis proving the asserted bounds is not yet available.