Cambridge English-Chinese Dictionary synonymizes "despite", before translating it". I understand how 管 means "influence".
without taking any notice of or being influenced by; not prevented by
Collins skips the intermediate step. p 317, ABC Chinese–English Comprehensive Dictionary (2003). I can't access the "slightly revised through July 2005" ed.
p 451, op. cit.
But what does 儘 signify here? What semantic notions underlie 儘 and "despite"?