Characterizations of the unit vector basis of $c_0$ or $\ell_p$ among minimal systems

Given a minimal system E for a Banach space X, Casazza, Dilworth, Kutzarova and Motakis introduced a multiplication $\otimes$ on span(E) and studied the Schauder bases for which $\|x \otimes y\|\approx \|x\|\|y\|$ asking whether every such basis is equivalent to the canonical unit vector basis of $c_0$ or $\ell_p$. We continue this study in the more general setting of minimal systems that are not assumed to be Schauder bases, and we obtain positive answers under hypotheses that are formally weaker than the ones considered so far: partial democracy instead of democracy or bidemocracy, and truncation quasi-greediness instead of quasi-greediness. Along the way we show that the above multiplicative estimate forces invariance under translations, we characterize when such a minimal system is a Schauder basis, and we give examples showing that the one-sided variants of the estimate behave very differently.