Linear-time computation of the tetra-separation decomposition

Let $G$ be a 4-connected finite graph with $n$ vertices and $m$ edges. The linear-time decomposition algorithm conjecture. There is an algorithm that returns the tetra-separation decomposition of $G$ from the main decomposition theorem in time

$$\mathcal{O}(n+m).$$

The claim is motivated by the existence of algorithms constructing 5-lean tree-decompositions of finite graphs in parameterized near-linear time. It remains an algorithmic conjecture because the source states that the proposed linear-time conversion to the tetra-separation decomposition is believed to exist, rather than proving it.