Stronger survival bound for symmetric-product contributions

Prove that the first degree M(n,k) in which the map \Sigma^{-2k}\tau_{>0}\mathbb{Sp}^{p^k}\to\Sigma^{-n}S^n(k) is nontrivial satisfies M(n,k)\geq c(n,k+1)-n for every k>0.

Background

The paper formulates a conjecture describing the homotopy groups of spheres in terms of the simpler approximations \hat S^n_k. A stronger form reduces this to a lower bound on the first nontrivial homotopy degree of a map involving the connected symmetric product spectrum. The known connectivity only gives the weaker bound M(n,k)\geq c(n,k)-n.