Novel theoretical constraints for color-octet scalar models *The work of OE was supported by the Agencia Estatal de Investigación (AEI, ES) and the European Regional Development Fund (ERDF, EU) (FPA2014-53631-C2-1-P, FPA2017-84445-P, and SEV-2014-0398). The work of CM was supported by the United States Department of Energy under Grant Contract DE-SC0012704. (September 2019)