Recursive formula for odd-primary Steenrod kernels

Prove or disprove the proposed recursive identity N_k=P^{p^{k-1}}N_{k-1}-\frac{1}{2}N_{k-1}P^{p^{k-1}} for the generator N_k of the first nontrivial kernel of right multiplication by P^{p^{k-1}}\cdots P^1\beta.

Background

The appendix constructs generators N_k for the first nontrivial kernels of right multiplication in the Steenrod algebra using stripping techniques. The authors obtain a complicated recursive formula and report that low-degree computations suggest a simpler relation, which they leave as an explicit question.