What is the longest slope $\gamma$ in the complement of Dehn surgery space of a cusped hyperbolic 3-manifold $M$? Here Dehn surgery space is the space of fillings such that the hyperbolic structure on the filling $M(\gamma)$ can be realized as a deformation of the original $M$.

This question is related to Ken Baker’s question:

Hyperbolic exceptional fillings of cusped hyperbolic 3-manifolds

However, Ken’s question is interested in the total number of slopes in this complement. This question is focusing on the longest such slope, where length is measured by displacement in boundary of a horoball packing (as in the set up for the $6$-theorem or $2\pi$-Theorem).

Of course, it is possible that this question as stated does not have a realizable answer, because there is no longest slope.

Here is a more carefully stated version:

What is the largest $L$ such that there exist a family of slopes $\gamma_i$ in manifolds $M_i$ such that $$\lim_{i \to \infty} length(\gamma_i) = L$$ and each $M_i(\gamma_i)$ is a hyperbolic manifold such that the hyperbolic structure cannot be realized as a deformation of the hyperbolic structure of $M_i$?

Of course, the fact that the 6-theorem is sharp implies that $L\geq 6$. Also the $2\pi$ Theorem says $L\leq 2\pi$.