Necessity of weak subordination for some strongly subordinated L\'evy processes

Let $\mathbf{X}$ be an $n$-dimensional L\'evy process and $\mathbf{T}$ be an $n$-dimensional subordinator. If $\mathbf{X}$ is a stacked independent L\'evy process and the components of $\mathbf{T}$ are indistinguishable within each stack, then the strongly subordinated process $\mathbf{X}\circ\mathbf{T}$ is a L\'evy process, otherwise it may not be. Weak subordination was introduced to extends strong subordination, always producing a L\'evy process $\mathbf{X}\odot\mathbf{T}$ even when strong subordination does not. Here, we prove that $(\mathbf{T},\mathbf{X}\circ\mathbf{T})$ is equal to law to $(\mathbf{T},\mathbf{X}\odot\mathbf{T})$ under the aforementioned conditions. In addition, we prove that if $(\mathbf{T},\mathbf{X}\circ\mathbf{T})$ is a L\'evy process, then it is necessarily equal to law to $(\mathbf{T},\mathbf{X}\odot\mathbf{T})$, in two cases: firstly, when $\mathbf{T}$ is deterministic and secondly, when $\mathbf{T}$ is pure-jump with finite activity.