McMullen's convex-core embedded-ball conjecture

Let $N$ be a hyperbolic $3$-manifold homotopy equivalent to a compact $3$-manifold $M$. Write $C(N)$ for the convex core of $N$.

McMullen's conjecture. The convex core $C(N)$ does not contain an embedded ball of radius $L$, where $L$ depends on the number of generators of $\pi_1(N)$.

This conjecture asserts a uniform restriction on the geometry of convex cores in terms of the rank of the fundamental group. The paper proves related uniform injectivity-radius bounds for books of $I$-bundles and acylindrical hyperbolizable $3$-manifolds, but the stated conjecture is not resolved here.