Finite-type representing measure conjecture for moment matrices with an involutive relation

Let $\beta$ be a moment sequence with a moment matrix $\mathcal M_2$ satisfying the column relation $\mathbb{Y}^2=\mathds 1$. A measure is said to be of type $(m_1,m_2)$, respectively of type $(m,1)$, when it has the corresponding finite type specified by these parameters.

Finite-type representing measure conjecture. The following are equivalent:

\begin{enumerate} \item $\mathcal{M}_2$ admits a measure. \item $\mathcal{M}_2$ admits a measure of type $(m_1,m_2)$, $m_1,m_2\in\mathbb{N}$. \item $\mathcal{M}_2$ admits a measure of type $(m,1)$, $m\in\mathbb{N}$. \end{enumerate}

The conjecture seeks a finite-type characterization of the existence of a representing measure under the relation $\mathbb{Y}^2=\mathds 1$. The preceding theorem proves a related equivalence for measures of types allowing zero parameters and shows constructively that, in a special case, a measure with one size-three atom can be replaced by one with a size-two atom; the full three-way equivalence remains open.