Does the shortest path have the correct order for some Hamiltonian path

Question

Given $G= \langle V,E\rangle$ such that $|V|=n$ and that $G$ have a Hamiltonian path from $v_1$ till $v_n$

Let $v_{l_1},v_{l_2}, \cdots , v_{l_n}$ be the shortest path from $v_1$ till $v_n$ so $v_{l_1}=v_1$ and $v_{l_n}= v_n$, is there $v_{h_1},\cdots , v_{h_n}$ some Hamiltonian path such that if $i<j$ then there is $a<b$ such that $v_{l_i} = v_{h_a}$ appears before $v_{l_j}=v_{h_b}$ ?

The question is simply that the order the nodes appears in the shortest path from $v_1$ to $v_n$ (not necessarily all the nodes since we are talking about shortest path) is preserved in at least one of the Hamiltonian paths from $v_1$ to $v_n$ ?

Answer 1

This graph seems to be a counterexample: