Generic finite convergence conjecture for generalized Moment-SOS hierarchies

Let $d_0$ and $d_i$ be positive degrees for $i\in [m]\cup\mathcal{E}\cup\mathcal{I}$. Consider polynomials $f\in\mathbb{R}[x]_{d_0}$, $a_i\in\mathbb{R}[x]_{d_i}$ for $i\in[m]$, a vector $b\in\mathbb{R}^m$, and $c_j\in\mathbb{R}[x]_{d_j}$ for $j\in\mathcal{E}\cup\mathcal{I}$. Write $a=(a_i)_{i\in[m]}$ and $c=(c_j)_{j\in\mathcal{E}\cup\mathcal{I}}$.

Generic finite convergence conjecture. There exists a finite set of polynomials $\varphi_1,\ldots,\varphi_L$ in the coefficients of $f$, $a$, $b$, and $c$ such that, if

$$\varphi_k(f,a,b,c)\neq 0,\qquad k=1,\ldots,L,$$

then the Moment-SOS hierarchy

$$--$$

has finite convergence.

The conjecture asserts that finite convergence holds outside the common zero set of finitely many coefficient polynomials, hence generically in the input data. The paper proves finite convergence under archimedeanness and optimality conditions, but the supplied text leaves open whether those conditions hold generically.