Necessity of transitivity for almost equivalence after horizontal Goodman surgery

Let $\phi^t$ be a pseudo-Anosov flow on a closed oriented $3$-manifold, let $c$ be a positive/negative horizontal surgery curve, and let $n$ be a positive/negative integer, respectively. Necessity conjecture. There exist such $\phi^t$, $c$, and $n$, with $\phi^t$ non-transitive, for which the surgically modified flow $\phi^t_{\frac{1}{n}}(c)$ is not almost equivalent to $\phi^t$. This would show that the transitivity hypothesis in the paper's almost-equivalence theorem cannot be removed; no example or resolution is supplied in the source.