Bochi–Katok–Rodriguez Hertz flexibility conjecture for Lyapunov exponents

Let $M$ be a $d$-dimensional manifold, let $m$ be a volume, and let $\operatorname{Diff}_m^{\infty}(M)$ denote the set of volume-preserving $C^{\infty}$ diffeomorphisms of $M$. Given a connected component $C \subset \operatorname{Diff}_m^{\infty}(M)$ and any list of numbers $\xi_1 \geq \cdots \geq \xi_d$ satisfying

$$\sum_{i=1}^{d} \xi_i = 0,$$

Bochi–Katok–Rodriguez Hertz flexibility conjecture. There exists an ergodic diffeomorphism $f \in C$ such that $\xi_i$, for $i=1,\ldots,d$, are the Lyapunov exponents of $f$ with respect to $m$. This predicts flexibility of the Lyapunov spectrum within every connected component of the space of smooth volume-preserving diffeomorphisms.