Does the entropy of systems with larger internal entanglement grow stronger?

It is known that when a system interacts with its environment, the entanglement contained in the system is redistributed since parts of the system entangle with the environment. On the other hand, the entanglement of a system with its environment is closely related to the entropy of the system. However, does this imply that the entropy of systems with larger internal entanglement will grow stronger? We study the issue using the simplest model as an example: a system of qubits interacts with the environment described by the quantum harmonic oscillator. The answer to the posed question is ambiguous. However, the study of the situation on average (using the simulation of a set of random states) reveals certain patterns and we can say that the answer is affirmative. At the same time, the choice of states satisfying certain conditions in some cases can change the dependence to the opposite. Additionally, we show that the entanglement depth also makes a small contribution to entropy growth.