Salamon's relation for Hochschild numbers of hyper-Kähler categories

Let $\mathcal{T}$ be a full admissible subcategory of the derived category of a smooth projective variety. Assume that $\mathcal{T}$ is hyper-Kähler of dimension $2r$, and write $\mathrm{hh}^k$ for its Hochschild cohomology numbers. Salamon's conjecture. Then

$$\sum_{j=1}^{2r}(-1)^j(3j^2-r)\,\mathrm{hh}^{2r-j}=\frac{r}{2}\mathrm{hh}^{2r}.$$

The formula is modeled on Salamon's relation for Betti numbers of hyper-Kähler manifolds; the source notes that it should first be checked for known hyper-Kähler categories and gives no general proof.