I wash silk, either by hand or in the machine on a delicate program. If I was worried I’d wash cold by hand, then roll it in towel and squeeze to remove the water, then dry flat.
it does depend on the construction though. If it’s lined or tailored with interfacing or anything like that, the different fabrics may react differently to the water, eg the lining may shrink slightly and then it won’t hang the same as before.
Similarly if it has pleats or similar, it may not come out as crisp as before.
the other potential issue is colour bleed if it’s patterned, eg a red pattern against white might bleed into the white. You reduce this risk by washing cold and quickly. You can also test for colour fastness.
Agree with the PP about the type and weave of the silk too. Some are more robust to washing than others.
Steam is a great way to freshen something that you are worried about washing. If you don’t have a steamer, Hang it in a steamy bathroom, then air it outside (out of direct sunlight), and that may be enough.