On the essential dimension of an algebraic group whose connected component is a torus

Let $p$ be a prime integer, $k$ be a $p$-closed field of characteristic $\neq p$,$T$beatorusdefinedover$k$,$F$beafinite$p$-group,and$1\to T \to G \to F \to 1$ be an exact sequence of algebraic groups. Extending earlier work of N. Karpenko and A. Merkurjev, R. L"otscher, M. MacDonald, A. Meyer, and the first author showed that $$\min \dim (V)-\min \dim (G)⩽\text{ed}(G;p)⩽\min \dim (W)-\dim (G),$$ where $V$ and $W$ range over the $p$-faithful and $p$-generically free $k$-representations of $G$, respectively. They conjectured that the upper bound is, in fact, sharp. This conjecture has remained open for some time. We prove it in the case, where $F$ is diagonalizable.