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We show that for every integer $k \geq 2$, the Res($k$) propositional proofsystem does not have the weak feasible disjunction property. Next, wegeneralize a recent result of Atserias and M\"uller [FOCS, 2019] to Res($k$).We show that if NP is not included in P (resp. QP, SUBEXP) then for everyinteger $k \geq 1$, Res($k$) is not automatable in polynomial (resp.quasi-polynomial, subexponential) time.

Failure of Feasible Disjunction Property for $k$-DNF Resolution and NP-hardness of Automating It