Strongly Koszul toric rings of 0–1 polytopes are trivial

Let $\mathcal{P}$ be a $(0,1)$-polytope, and let $k[\mathcal{P}]$ be the toric ring generated by $\mathcal{P}$. A ring $R$ is trivial if it can be constructed from polynomial rings by repeatedly applying tensor products and Segre products. The authors' conjecture. If $k[\mathcal{P}]$ is strongly Koszul, then $k[\mathcal{P}]$ is trivial. The conjecture seeks a structural characterization of strongly Koszul toric rings arising from $(0,1)$-polytopes; no resolution is supplied in the statement provided.