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A famous conjecture of Parkin-Shanks predicts that $p(n)$ is odd with density$1/2$. Despite the remarkable amount of work of the last several decades,however, even showing this density is positive seems out of reach. In a 2018paper with Judge, we introduced a different approach and conjectured the"striking" fact that, if for any $A \equiv \pm 1\ (\bmod 6)$ the multipartitionfunction $p_A(n)$ has positive odd density, then so does $p(n)$. Similarly, thepositive odd density of any $p_{A}(n)$ with $A\equiv 3\ (\bmod 6)$ would implythat of $p_3(n)$.  Our conjecture was shown to be a corollary of an earlier conjecture of thesame paper. In this brief note, we provide an unconditional proof of it. Animportant tool will be Chen's recent breakthrough on a special case of ourearlier conjecture.

Deducing the positive odd density of $p(n)$ from that of a multipartition function: An unconditional proof