The newly-minted trillionaire could use their wealth to monopolize positional goods: social status, housing, politicians' attention. This could worsen your standard of living if you compete with them for the same positional goods.
You define "standard of living" positionally, which is not how it is usually defined, but does make a lot of sense given the prevalence of envy. By that definition, yes, inequality does absolutely worsen standard of living. Points for being honest about your motivations, unlike every other person who rails against inequality.