Closedness of the orbit-closed \(C\)-numerical range and submajorization

For a positive trace-class operator \(C\) and a bounded operator \(A\), we provide an explicit description of the closure of the orbit-closed \(C\)-numerical range of \(A\) in terms of those operators submajorized by \(C\) and the essential numerical range of \(A\). This generalizes and subsumes recent work of Chan, Li and Poon for the \(k\)-numerical range, as well as some of our own previous work on the orbit-closed \(C\)-numerical range.