Non-normative logical pluralism and the revenge of the normativity objection

Logical pluralism is the view that there is more than one correct logic. Most logical pluralists think that logic is normative in the sense that you make a mistake if you accept the premisses of a valid argument but reject its conclusion. Some authors have argued that this combination is self-undermining: Suppose that L1 and L2 are correct logics that coincide except for the argument from Γ to ϕ, which is valid in L1 but invalid in L2⁠. If you accept all sentences in Γ, then, by normativity, you make a mistake if you reject ϕ. In order to avoid mistakes, you should accept ϕ or suspend judgment about ϕ. Both options are problematic for pluralism. Can pluralists avoid this worry by rejecting the normativity of logic? I argue that they cannot. All else being equal, the argument goes through even if logic is not normative.