With their article "Restoring indigenous names in taxonomy", Gillman & Wright (2020) initiated a discussion that culminated in a set of proposals to amend the International Code of Nomenclature for algae, fungi, and plants (ICN) by allowing the retroactive replacement of scientific names and/or epithets of plants and other organisms in favor of "indigenous names" (vernacular names used by Indigenous Peoples). Here I emphasize several main topics and questions critically important for the suggested implementation of the proposals by Wright and Gillman. My conclusions regarding the proposals by Wright and Gillman are as follows. The "definition" of Indigenous People proposed by Wright and Gillman for the ICN is dramatically different from the working definitions and identification/self-identification criteria of Indigenous Peoples provided in the UN Declaration on the rights of Indigenous Peoples (United Nations, 2007) and other relevant international documents. That "definition" is vague, potentially discriminatory, may bring unnecessary conflicts, and cannot be used in practice. Nomenclatural committees and the taxonomic community as a whole should not be involved in political debates about which peoples are indigenous and which are not, which bodies represent those peoples and which do not have that right. Giving special privileges in nomenclature to some groups of Indigenous Peoples but not to other peoples with comparable levels of traditional knowledge would be an act of unexplainable and unjustified discrimination. The argument of "demonstrable chronological priority" of vernacular "indigenous" names over scientific names used by Wright and Gillman is irrelevant in biological nomenclature and taxonomy and cannot be used in practice. The proposal to allow the use of "indigenous" diacritics in biological nomenclature would result in allowance of dozens diacritic signs actually used by various Indigenous ethnic groups, and should not be accepted. The replacement procedure proposed by Wright and Gillman will not work as they intended because they failed to outline workable rules for it. Science (the scientific nomenclature of organisms included) should not be biased or distorted politically, racially, ethnically, socially, genderly, religiously or otherwise. Any attempts or proposals aimed at granting preferences in biological nomenclature to any political, racial, ethnic, social, gender, religious or other group or groups should be rejected as discriminatory acts. Consequently, the "proposals" by Wright and Gillman should be rejected by the international community of taxonomists of plants, fungi, and algae.