A Variance–AM–GM Inequality Conjecture: A Call to the Mathematical Community

 This note proposes a new inequality linking the classical arithmetic and geometric
 means with the variance of positive real variables. The conjectured relation, which
 we call the Variance–AM–GM Inequality, offers a quantitative bridge between additive
 and multiplicative dispersion. Its resolution may yield new insights in convex analysis,
 information theory, and statistical physics. The purpose of this paper is to formally
 state the conjecture, provide motivation for its structure, and invite the mathematical
 community to verify, refine, or refute it. 

*** UPDATE: Please note two followups have been released

https://doi.org/10.5281/zenodo.17345975 and https://doi.org/10.5281/zenodo.17345993 

The conjecture was partial flawed however we were able to build further upon AM-GM