A $4$-vertex-colored planar graph $G$ is a planar graph $G \overset{\text{def}}{=} (V, E, C)$ where $V$ and $E$ are as usual and $C$ consists of pairs $(v \in V, c \in \{1,\dots,4\})$ such that $\forall (v_1, c_1), (v_2, c_2) \in C$, $v_1 \neq v_2$ and if $c_1 = c_2$, $(v_1, v_2) \notin E$.
A Kempe-chain $K_{c_1,c_2}$, $c_i \in \{1,\dots,4\}, c_i \neq c_j$ is the connected induced subgraph containing a vertex $v$ such that $(v, c_i) \in C$ and closed under the addition of vertices $v_1$ such that $v_2 \in V(K), (v_1, v_2) \in E(G)$ and $(v_1, c_i) \in C(G)$.
A Kempe switch is an operation that recolors the vertices within a given Kempe chain $K_{c_1,c_2}$ such that $(v, c_1) \mapsto (v, c_2)$ and $(v, c_2) \mapsto (v, c_1)$. Kempe switching is significant because it preserves the correctness of the coloring and is the only such operation known to do so.
My question is this - are the set of all proper $4$-vertex-colorings of a given planar graph $G$ closed under Kempe-switching? I.e. is it always possible to transform one vertex-coloring into another by a sequence of Kempe switches.
I'd much appreciate a reference to a proof if one exists. If not, how significant is this problem in the search for a non-computational proof of the 4CT?
No. Have a look at this paper of Mohar which gives an infinite family of planar triangulations with 4-colourings that are only Kempe-equivalent to themselves.