Translation:The granddaughters visit their grandparents.
Even in this sentence when there is no other 3rd person subject, you use “siajn”? I would have thought it was “iliajn”.
Yes. If you said "iliajn" it would mean that they visit someone elses grandparents.
The granddaughters (Alice and June) visit their (Isabell and Amandas) grandparents.
The granddaughters (Alice and June) visit their (own) grandparents.
But the thing is, you should never run into problems with those being mixed up. Because you never would have introduced Isabell and Amanda. Alice and June would've been introduced in an earlier sentence but there is no reason that there would've been a conflict with just the one kind of "their." But I guess if that's just a rule in Esperanto I should accept it. I'm just saying that it's unnessesarry
Could you expand a bit on why this would never be a problem? (Not that a problem is a requirement for making a language stronger, but I'd like to know the argument you're trying to make.)
So "siajn" is for our own grandparents in this case and "iliajn" is for someone else? Does it right?
No, siajn is referring to the subject of the sentence. Iliajn would be someone else's parents.
Why? The ambiguity on this sxiajn / ilijn thing isn't an issue in any other language I know, including English.
Furthermore, 'sxiajn' could still be referring to a 3rd person singular (like Jack and Jill visiting Molly's parents) so the ambiguity remains nonetheless.
It is not ŝiajn but siajn. There is no ambiguity. It is a universal reflexive pronoun that always refers back to the subject.
Because they are visiting their own grandparents.
Had it been ŝiajn, it would have been about "ŝia" (her) grandparents. The context would have to explain who this other female is that the sentence would be talking about in that case.
Can "siajn" be singular? Is there a word to refer to one person of an unspecified gender like "them" in English?
Yes, it would be "sian" or "sia." And yes, it is "gxi," which would be rude to call slmeone in engish "here it is," but is it fine in Espersnto-"Jun estas gxi"
Mi estas komencanto, but it seems like it's "siajn" because both grandchildren are female ("la nepinoj vizitas siajn geavojn.") If both grandchildren were male ("La nepoj...") then the gender of the possessive adjective would be male ("...vizitas liajn geavojn.") If it were merely grandchildren ("la genepoj...") then the gender of the possessive adjective would be inconsequential ("...vizitas iliajn geavojn"). Cxu ne?
I presume you've figured out why this is wrong by now: It is not ŝiajn, but siajn. Ŝiajn would always be wrong, because it is singular and nepinoj is plural. Si is a universal reflexive pronoun that refers to the subject, regardless of gender or number.
La nepino vizitas siajn geavojn: the granddaughter visits her (own) grandparents. La nepino vizitas ŝiajn geavojn: the granddaughter visits her (someone else's) grandparents. La genepoj vizitas siajn geavojn: the grandchildren visits their (own) grandparents. La genepoj vizitas iliajn geavojn: the grandchildren visits their (someone else's) grandparents. etc.
The English way of doing it is potentially ambiguous, and the Esperanto way of doing it is never ambiguous – unless you use the wrong pronoun, of course.
That said, if there is nothing for the wrong pronoun to refer back to if you wrongly use ŝiajn or iliajn, it would be interpreted simply as a mistake, and the meaning would still be clear.