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 $\neq p$, $T$ be a torus defined over $k$, $F$ be a finite $p$-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 $$\min\dim(V) - \min\dim(G) \leqslant \text{ed}(G; p) \leqslant \min \dim(W) - \dim(G),$$ 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.