 Solution

Every finitely presented group $P$ embeds in $P\times C_2$ by $p\mapsto (p,1)$. This direct product of groups is finitely presented: combine finite presentations of the factors and add the finitely many cross-commutation relations between their generators. Projection onto $C_2$ gives a nontrivial finite quotient of a group.

Thus no finitely presented $P$ can serve as the forbidden witness $A_{-}$ for this property. Hence