Odd-order and normal rank two for all primes implies Sylow tower

This article states and (possibly) proves a fact that is true for certain kinds of odd-order groups. The analogous statement is not in general true for groups of even order.
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This article describes a result about finite groups where the isomorphism types of the Sylow subgroup (?)s implies certain properties of the whole group. Note that all $p$-Sylow subgroups for a given prime $p$ are conjugate and hence are isomorphic, so the statement makes sense.
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Statement

Suppose $G$ is a finite group of odd order. Further, suppose that for every prime $p$ dividing the order of $G$, and every $p$-Sylow subgroup $P$ of $G$, the normal rank of $P$ is at most two. Then, $G$ has a Sylow tower.

Related facts

Breakdown for even order

The result fails if we allow groups of even order. The smallest counterexample is the symmetric group of degree four, where both Sylow subgroups have normal rank at most two, but the group does not possess a Sylow tower.