learning experience
Much of the feedback so far weighs in on the side of "required -- family is family". Someone cited honoring one's parents (the source of the request), and a couple people mentioned protecting a life (the sibling is apparently in real danger of injury without someone there).
I, on the other hand, am leaning toward "forbidden", though "permitted" is a possibility. Definitely not "required", though.
The issue is complex. While the sibling needs a caregiver, that's a service that can be hired -- so there's no apparent need for the poster to do it personally. Of course it's important to honor one's parents (this comes up a lot in text), but the talmud also teaches that if a parent asks you to transgress the Torah, you must decline (Bava Metzia 32a). This raises the question of whether attending another religion's worship service -- on its second-holiest day, to boot -- is avodah zara, forbidden worship. Is it enough if you don't intend to worship? What if you don't participate? What if you don't listen? That is a complex question with varied answers depending on circumstances, ranging from exactly what will take place to the strength of your own Jewish education and commitment, and you really need to ask your rabbi for a personal ruling.
I think the experience of facing this issue is valuable for the conversion candidate, actually. As a member of a minority religion (that sometimes faces hostility from others), sometimes you are going to have to make choices between your religion and your family/friends/society -- things like this, or resolving Shabbat issues with your employer, or various other matters. Finding out how you will handle those choices before it's "too late" -- before you convert and acquire new obligations -- seems useful to me.
I assume that most conversion candidates face some sort of religion-vs-world-at-large test during the process, but I don't actually know.

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However, I'd not point to genuflecting as a problem; we have our own version, after all. And, unfortunately, one can stretch a point (possibly too far) to claim that we've invested perhaps a bit more iconography in Torah scrolls than is justified by the need to show proper respect for them. (On the other hand, we're not perfect either; maybe we need to do some self-examination on that issue.)
I will agree that the whole business with the cross is unjustifiable iconography, as is the "fish" that is increasingly common, and other icons used by various denominations (many Protestant branches use an icon of flame with a cross motif worked in, and then there's the Catholic "sacred heart").
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Also, afaik, Greek Orthodox is the Eastern church that ignores Rome, and Byzantine Catholic is the Eastern church that follows Rome. It would seem that the former should be the less influenced of the two, being, I'd guess, made up of those broken-off factions you mentioned, if they still exist. (It seems to make no sense that the Romans would just let half the Eastern church leave if they had managed to take them over when the Byzantine Empire fell.)
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