This is not an answer. Recently, I have been thinking questions related to yours, and I posted some of them on the board, so you may have seen them. I also posted related references in my comments there, but I think that you have already read those papers. It’s possible to ask a more general question: “Which subsets $S\subset\mathbb{B}(\mathcal{H})_{sa}$ have the property $S^m=S_m=S$?” or “Which real linear subspaces $S\subset\mathbb{B}(\mathcal{H})_{sa}$ have the property $S^m=S$?” It would be interesting if these simple conditions implied $S$ being strongly closed.

This is not an answer. Recently, I have been thinking questions related to yours, and I posted some of them on the board, so you may have seen them. I also posted related references in my comments there, but I think that you have already read those papers. It’s possible to ask a more general question: “Which subsets $S\subset\mathbb{B}(\mathcal{H})_{sa}$ have the property $S^m=S_m=S$?” or “Which real linear subspaces $S\subset\mathbb{B}(\mathcal{H})_{sa}$ have the property $S^m=S$?” It would be interesting if these simple conditions implied $S$ being strongly closed.