Let $H$ be a separable Hilbert space over $\mathbb{C}$, say $\ell_2$ for simplicity. Let $\mathcal{K}(H)$ denote the space of all compact operators on $H$ and $\mathcal{P}(H)$ the set of all finite rank orthogonal projections on $H$ (so $\mathcal{P}(H)\subset\mathcal{K}(H)$). Assume that $(x_n^*)_{n\in\mathbb{N}}$ is a sequence of bounded functionals on $\mathcal{K}(H)$ such that $\lim_{n\to\infty}x_n^*(P)=0$ for every $P\in\mathcal{P}(H)$.

**Question:** Is it true that $\lim_{n\to\infty}x_n^*(T)=0$ for every $T\in\mathcal{K}(H)$?

Equivalently, is $(x_n^*)_{n\in\mathbb{N}}$ norm bounded? Such a situation holds e.g. in von Neumann algebras (a result due to Darst '67) or C*-algebras of the form $C(K)$ where $K$ is the Stone space of a $\sigma$-complete Boolean algebra (Nikodym '33).