Family Heirloom Pearls (purchased in Japan in the late '40s)

[Soaring Bean]

Hello! Any help you can provide would be greatly appreciated! We used to have two of these necklaces, but now we only have one. I would love to figure out anything about these pearls, and to see if it's possible to replace the one that was lost.

1. Do the pearls feel slightly gritty or smooth when rubbed gently against another pearl? Edited to add: I have stopped recommending rubbing pearls against teeth (“Tooth test”) Teeth are harder than pearls and can scratch them.
If they are gritty, they are likely to be real nacre (genuine pearls.) If they feel smooth, they are likely imitation pearls.

They feel pretty smooth, to be honest.

2. Please provide clear, in-focus photos without flash against a white background (a paper towel works nicely.)
Include close-ups of the clasp (front and back) and a few of the pearls. If there are flaws, include a photo of those. Also the box they came in, and tags if you have them.

I note no flaws in the pearls themselves. The thread seems to not be in the best of shape (seems a bit frayed). I don't have any tags, but the box is navy blue, seemingly made of plastic.

3. Any history you can give us about the pearls. Where/when you or your relative got them, any documentation you have (receipts, appraisals), their price range if you know it, etc.

My grandfather purchased two necklaces while he was stationed in Japan, sometime after WWII. I do not know the price, but I know my grandfather was the son of sharecroppers, so he wouldn't have had much money to spare. When my grandmother passed away, she gave them to my mother and aunt.
My aunt has had her strand restrung. I am not certain whether the photo is of her necklace, or the one my mother received.

4. Describe any marks on the clasp. These may be numbers (14K, 585, 750, 925 etc.) or brand names or even pictures. If the clasp has stones, have you had them tested, or do you have documentation about what they are?

The clasp has 585 on the back, and seems to have a pearl on the front?

5. Measure the pearls, with a millimeter ruler if possible. If they are graduated, measure the largest and smallest pearls.

The largest pearl appears to be 7-8mm, and the next three on each side are ~6mm. At the midpoint in the string, they're 4mm, and at the end they're down to 2-3mm.
The pearls-only strand is 42.5cm, or a bit less than 17in