The Integral Conjecture for polynomial moment subspaces

Let $B\subset\mathbb R$ be open and let $\sigma$ be a positive measure such that $\int_B g(t)\,d\sigma$ exists and is finite for every $g\in\mathbb C[t]$. Define

$$V_B(\sigma)=\left\{f\in\mathbb C[t]\mid \int_B f\,d\sigma=0\right\}.$$

Integral conjecture. The subspace $V_B(\sigma)$ is a Mathieu subspace of $\mathbb C[t]$. This is one of the one-dimensional Integral and Image Conjectures; the source records special cases but gives no general resolution.