Let $H$ be a separable infinite-dimensional real Hilbert space. We consider operators in $H.$

Nuclear norm of a nuclear operator is the sum of its singular values. A nuclear, positive and self-adjoint operator is called S-operator.

Does the following criterion hold true?

A sequence $A_n$ of S-operators converges in nuclear norm to S-operator $A$ if, and only if, $A_n$ weakly converges to $A$ and there exists an orthobasis $e_k$ such that the series $\Sigma_{k=1}^{\infty}(A_n e_k, e_k)$ converges uniformly in $n=1, 2\ldots$

I can prove the necessity only. But does the sufficiency hold true as well?