Let $π\in \mathfrak{S}_m$ and $σ\in \mathfrak{S}_n$ be permutations. An occurrence of $π$ in $σ$ as a consecutive pattern is a subsequence $σ_i σ_{i+1} \cdots σ_{i+m-1}$ of $σ$ with the same order relations as $π$. We say that patterns $π, τ\in \mathfrak{S}_m$ are strongly c-Wilf equivalent if for all $n$ and $k$, the number of permutations in $\mathfrak{S}_n$ with exactly $k$ occurrences of $π$ as a consecutive pattern is the same as for $τ$. In 2018, Dwyer and Elizalde conjectured (generalizing a conjecture of Elizalde from 2012) that if $π, τ\in \mathfrak{S}_m$ are strongly c-Wilf equivalent, then $(τ_1, τ_m)$ is equal to one of $(π_1, π_m)$, $(π_m, π_1)$, $(m+1 - π_1, m+1-π_m)$, or $(m+1 - π_m, m+1 - π_1)$. We prove this conjecture using the cluster method introduced by Goulden and Jackson in 1979, which Dwyer and Elizalde previously applied to prove that $|π_1 - π_m| = |τ_1 - τ_m|$. A consequence of our result is the full classification of c-Wilf equivalence for a special class of permutations, the non-overlapping permutations. Our approach uses analytic methods to approximate the number of linear extensions of the "cluster posets" of Elizalde and Noy.