Similarity conjecture for cubic rational helical space curves

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be cubic rational helical space curves with proportionality constants $\mu_1$ and $\mu_2$, respectively. The curves have proportionality constants of equal modulus when $|\mu_1|=|\mu_2|$. Similarity conjecture. Any two cubic rational helical space curves with proportionality constants of equal modulus are similar. This conjecture asserts that, for helical cubics, the necessary condition from the similarity proposition is also sufficient; the paper reports no counterexample among the examples tested, while the general converse fails for helical quintics.