Spiral-knot parameter uniqueness conjecture

Let $S(p,q,\varepsilon)$ and $S(p',q',\varepsilon')$ be spiral knots, with $p,q,p',q'$ and the signs $\varepsilon,\varepsilon'$ as in the spiral-knot notation. Parameter uniqueness conjecture. If

$$S(p,q,\varepsilon)=S(p',q',\varepsilon'),$$

then either $p=p'$ and $q=q'$, or $S(p,q,\varepsilon)$ is a torus knot, in which case $p=q'$ and $q=p'$. This conjecture seeks to strengthen the known conditions under which two spiral knots are equal; its connection with the meridional rank conjecture provides additional motivation, but the claim itself remains open.