Suppose $X$ is an $n$-dimensional regular CW-space (a space with a regular CW decomposition). What are the weakest sufficient conditions required on $X$ to ensure that all regular CW decompositions $(X_{i})$ of $X$ are such that $X_{i} \setminus X_{i-2}$ is path-connected for all $1 \leq i \leq n$?
It is common knowledge that $X$ being connected gives us that $X_{1} \setminus X_{-1} = X_{1}$ is path-connected.
If we identify two disks at a point $x$ on their boundary and then give them the following decomposition,
we have that $X_{2} \setminus X_{0}$ is not path-connected.
In this case the obstruction is caused by the existence of the singularity $x$.
To me it seems like the two ways to remove such a singularity are simply require that $X$ is a manifold rather than a pseudomanifold, or perhaps requiring that the space has no boundary? Are either of these directions viable conditions? Is there a weaker condition (clearly when $X$ is a $1$-dimensional regular CW complex connectedness is sufficient)?
(EDIT): Sketch proof that $X$ is a manifold is sufficient: We have the base case that $X_{1} \setminus X_{-1} = X_{1}$ is path-connected by a folk theorem. Note that connectedness and path-connectedness are identical for manifolds.
Suppose that $X_{k} \setminus X_{k-2}$ is path-connected where $k\geq 2$.
Consider $X_{k+1} \setminus X_{k-1}$ and let $A$ and $B$ be $(k+1)$-cells, it is sufficient to find a path between them which lies entirely inside $X_{k+1} \setminus X_{k-1}$. As both $A$ and $B$ have boundaries in $X_{k}$ there exists $a \in \partial A$ and $b \in \partial B$ such that $a$ and $b$ are in $X_{k} \setminus X_{k-1}$.
Let $\gamma$ be a path from $a$ through $X_{k} \setminus X_{k-2}$ to $b$, such a path exists by the induction hypothesis. Suppose that $\gamma$ contains elements of $X_{k-2}$, that is; it is not a path through $X_{k+1} \setminus X_{k-1}$.
We aim to amend $\gamma$ so that we avoid elements of $X_{k-1}$. In the two dimensional case this amendment can be illustrated as follows:
where the blue dotted line is $\gamma$ and green curves show the amendments. We may do this as, if $X$ is a surface and $p \in \gamma \cap X_{0}$, there is a small neighborhood $U$ around $p$ homeomorphic to an open ball in $\mathbb{R}^{n}$. We have that $\partial U$ intersects $\gamma$, is contained in $X_{2} \setminus X_{0}$ and is path-connected so we may use it to traverse around $p$.
In general $\gamma \cap X_{k-1}$ may not be discrete singletons like in the case when $k$ is $1$, but in a similar manner(?): if $C$ is a connected component of $\gamma \cap X_{k-1}$ we assign a small neighborhood $U_{t}$ to each $t \in C$ so that the union $$U_{C} = \bigcup_{t \in C} U_{t}$$ is an open cover of $C$ and if we pick $U_{t}$ sufficiently small we may use $\partial U_{C}$ to traverse around $C$ while remaining in $X_{k+1} \setminus X_{k-1}$.