Colliot-Thélène–Karpenko–Merkurjev conjecture on essential dimension of projective modules

Let $k$ be a field, let $d\geq 1$, and let $D$ be a central division algebra of degree $d$ over $k$. For a positive rational number $r$, write $\operatorname{Mod}_{D,r}$ for the functor assigning to each field extension of $k$ the isomorphism classes of projective $D$-modules of rank $r$. Colliot-Thélène–Karpenko–Merkurjev conjecture.

$$\operatorname{ed}_k\bigl(\operatorname{Mod}_{D,1/d}\bigr)=\sum_{p\mid d}\bigl(p^{v_p(d)}-1\bigr),$$

where the sum is over all primes $p$. Equivalently, this predicts the canonical dimension of the Severi–Brauer variety associated with $D$ in terms of the prime-power factors of $d$. The conjecture is attributed to Colliot-Thélène, Karpenko, and Merkurjev and is used to determine essential dimensions of quiver-representation functors; its resolution status is not specified in the source.