The equality conjecture for iterated local dimension sets

Let $X$ be a topological space that is almost zero-dimensional, meaning that it has a basis of open sets whose boundaries are zero-dimensional. Define

$$\Lambda(X)=\{x\in X:X\text{ is 1-dimensional at }x\}$$

and

$$\Lambda^2(X)=\Lambda(\Lambda(X))=\{x\in\Lambda(X):\Lambda(X)\text{ is 1-dimensional at }x\}.$$

Equality conjecture. If $X$ is almost zero-dimensional, then

$$\Lambda^2(X)=\Lambda(X).$$

The preceding result establishes only that $\Lambda^2(X)$ is dense in $\Lambda(X)$ for almost zero-dimensional spaces; the conjecture asks whether this dense subset must equal all of $\Lambda(X)$.