Let $(X, \mathcal{T})$ be a topological space, and $S(X)$ be the set of topologies on $X$. Then
$$\mathcal{T}^P = \bigcap \{ \mathcal{T}^* \in S(X): \mathcal{T} \subset \mathcal{T}^* \land \mathcal{T}^* \text{ is locally connected}\}.$$
is the smallest locally connected topology which contains $\mathcal{T}$; a simpler direct proof here.
I'm finding it somewhat difficult to work with this definition in my research.
Problem
Give a useful alternative characterization of the topology $\mathcal{T}^P$.
Background
The first link characterizes $\mathcal{T}^P$ as a fixed point of a transfinite sequence. But that definition also seems difficult to work with.
Consider the following (different) construction for sum-connected topologies. It can be shown that the smallest sum-connected topology (defined as connected components being open) is given by
$$\mathcal{T}^+ = \bigcap \{ \mathcal{T}^* \in S(X): \mathcal{T} \cup \mathcal{C}(\mathcal{T}^*) \subset \mathcal{T}^*\},$$
where $\mathcal{C}(\mathcal{T}^*)$ are the connected components of $\mathcal{T}^*$. This is the topology generated by $\mathcal{T} \cup \mathcal{C}(\mathcal{T})$. It can also be shown that
$$\mathcal{T}^+ = \{U \subset X : \forall C \in \mathcal{C}(\mathcal{T}): U \cap C \in \mathcal{T}|C\}.$$
That latter $\mathcal{C}(\mathcal{T})$-coherent topology is simpler to work in most cases. Here $\mathcal{T}|C$ is the subspace topology of $\mathcal{T}$ on $C$.
In analogy it would be nice if $\mathcal{T}^P$ could also be given as some simple coherent topology.
A space is locally connected if and only if each open subspace is sum-connected. So perhaps $\mathcal{T}^P$ could be given in terms of $\mathcal{T}^+$ somehow. When I tried it, I was driven towards a similar transfinite induction as in the first link. But perhaps there is some another way.
