The silk may well be fragile after this amount of time, and I think it depends on whether you want to seek professional advice or not.
You could begin with a fine powder - fullers' earth, cornflour or talc. Sprinkle on and use a brush (small paintbrush, cosmetic brush, baby's hairbrush) to work it in a little bit, and leave for an hour or two. Then upend the box and shake the powder out, use a brush to help. You can also use a small handheld vacuum on lowest setting. That may 'lift' it enough.
I have a similar box, but it was actually a lady's cosmetic box originally. It is leather & velvet lined.