Hamiltonian path pattern conjecture for labelled graph notations

Let $G$ be a graph with labelled linear notation $I[G]$, where the colouring algorithm assigns natural-number colours $1,2,\ldots,n$ to the vertices and adjacent numbers must be connected by an edge. A Hamiltonian path is a path that passes through every vertex exactly once. The notation $\#\ \#\ \#$ denotes a string of return codes with abstract edges $\rightarrow \#1\ldots\rightarrow \#m$.

Hamiltonian path pattern conjecture. In order for a Hamiltonian path to exist on $I[G]$, it is necessary and sufficient that $I[G]$ satisfy one of the four possible patterns:

$$I[G] = 1 \Big[_{1} \rightarrow k \Big[_{2} \ldots \Big] \, \Big]$$ $$I[G] = 1 \Big[_{1} \rightarrow k_{1} \Big[_{2} \ldots \Big]; \rightarrow k_{2} \Big[_{2} \ldots \Big] \, \Big]$$ $$I[G] = 1 \Big[_{1}... \rightarrow k_{1} \Big[_{i1} \#\#\#; \rightarrow k_{2} \Big[_{i2} \rightarrow \#1... \Big] \, \Big]; \rightarrow k_{3} \Big[_{i3} \ldots \Big] \, \Big]$$ $$I[G] = 1 \Big[_{1}... \rightarrow k_{1} \Big[_{i1} \#\#\#; \rightarrow k_{2} \Big[_{i2}... \rightarrow k_{3} \Big[_{i3} \rightarrow \#1; \, \#\#\# \Big] \, \Big] \, \Big]; \rightarrow k_{4} \Big[_{i4} \ldots \Big] \, \Big]$$

Inside the blocks $\Big[_{2}\ldots \Big]$, $\Big[_{i2}\ldots \Big]$, $\Big[_{i3}\ldots \Big]$ and $\Big[_{i4}\ldots \Big]$, the number of nested brackets must be exactly equal to the number of abstract vertices, with no extra branching allowed. This conjecture was put forward based on an enumeration of graphs with at most seven vertices; its validity beyond that tested range remains open.