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Source. Section 4.15 and its extension, printed pp. 458–459 (published PDF).

Statement. Let $P$ be a nonempty connected subgraph of $G$. Suppose $M$ is a matching whose internal edges leave exactly one vertex of $P$ uncovered, $M/P$ is maximum in $G/P$, and some stem for $M/P$ has the contracted vertex $p=P/P$ as its tip. Then $M$ is maximum in $G$.

The paper says only "a subgraph"; connectedness is the condition under which Section 4.9 (pp. 456–457) defines shrinking, so it is stated here.

For pairwise disjoint connected subgraphs $P_1,\ldots,P_s$, suppose each internal restriction is near-perfect and the quotient restriction of $M$ is maximum. If all quotient vertices $p_i$ are outer vertices of one planted tree, then $M$ is maximum before all these contractions.

Proof. The stem ending at $p$ lifts outside $P$ to a stem ending at its single internally unmatched vertex: if the stem has an edge at $p$, that matching edge must attach at this vertex. If the stem has length zero, $p$ is exposed, so the internally unmatched vertex is exposed in $G$ too.

Flip this stem in $M$. The resulting matching $M'$ has the same cardinality as $M$, and its single exposed vertex in $P$ is now also exposed in the ambient graph. Its quotient has the same size as $M/P$ and is therefore maximum.

If $M'$ were not maximum, take an augmenting path using Section 3.7. If it avoids $P$, it is already an augmenting path in the quotient. Otherwise at least one of its two exposed endpoints lies outside $P$. Start there and stop at the first vertex of $P$. The entering edge is nonmatching, since no edge of $M'$ crosses $P$. After contraction this segment is an alternating path from the original exposed endpoint to the exposed vertex $p$, again augmenting the quotient. Both cases contradict its maximal cardinality. Thus $M'$ and $M$ are maximum.

For the second statement, choose among the still-contracted $p_i$ one of greatest distance from the planted-tree root. Its root path is a stem. Expand this $P_i$ while retaining the given internal restriction of $M$. The first statement proves that the new matching is still maximum.

No root path to another remaining $p_j$ passed through $p_i$: if it did, $p_j$ would have greater distance. Those paths and their matching edges therefore survive unchanged as stems after the expansion. Repeat in decreasing root depth. At each step the single-contraction argument applies, and eventually all subgraphs have been restored. $\square$

This expands the source's ordering argument. It does not require the subgraphs to be arbitrary factor-critical graphs; their given near-perfect restrictions and the actual stems are the hypotheses used here.