Nested matching criterion for a regular subgraph

Let $p$ be prime and set $s=2p-2$. If a bipartite graph has edge-disjoint perfect matchings $M_0,\dots ,M_{\sigma -1}$ on nested nonempty vertex sets, and $|V(M_{i+s})|>s|V(M_i)|/(s+1)$, then the union of $s+1$ consecutive matchings in a graph has maximum degree at most $2p-1$ and more than $(p-1)|V(M_i)|$ edges. The prime-regular subgraph from Boolean polynomial constraints gives a $p$-regular subgraph. Its perfect matching decomposition gives a $k$-regular subgraph for every $k\le p$. If none exists, every $s$ steps shrink the vertex set by factor at most $s/(s+1)$, bounding $\sigma$ by a constant times $\log n$.