For others searching: https://www.scotusblog.com/cases/case-files/united-states-v-skrmetti/ has all, and I mean all, the files.
In the table, under "Opinion", you get all the justices' opinions in a huge, over 100 page file. Justice Alito's opinion starts on page 64. In page 83 he writes "The parties in this case also admit that transgender status is not an immutable characteristic. See Tr. of Oral Arg. 97-98. Instead, a person’s gender identity may "shif[t]," and a person who is transgender now may not be transgender later." The bolding is mine because my next paragraph follows that reference.
In the Scotusblog table, under "Argument", there's a link to an audio recording of the argument in court and a transcript of the audio recording. The transcript pages 97-98 feature Justice Alito getting activist barrister Chase Strangio to admit in court that "trans" includes gender fluidity and that detransitioners exist, therefore being trans isn't immutable and so laws that might vary in their effect based on whether someone is trans shouldn't be subject to the level of scrutiny[1] that laws disproportionately affecting Black people should be.
JUSTICE ALITO: Are there individuals who are born male, assigned male at birth, who at one point identify as female but then later come to identify as male, and, likewise, for individuals who are assigned female at birth, at some point identify as male - as female - I'm sorry - identify as male but later come to identify as female? Are there not such people?
MR. [sic] STRANGIO:There are such people. I agree with that, Justice Alito.
JUSTICE ALITO: So it's not an immutable characteristic, is it?
Back to Justice Alito's opinion:
Moreover, transgender status, unlike race and sex, is often not accompanied by visibly identifiable characteristics. A person's "gender identity" is an "internal sense," Merriam-Webster’s Collegiate Dictionary, at 520, and transgender persons as a class do not uniformly "exhibit obvious, immutable, or distinguishing characteristics that define them as a discrete group," Lyng, 477 U. S., at 638. Nor do they necessarily tend to "carry an obvious badge" of their membership in the class that might serve to exacerbate discrimination. Mathews, 427 U. S., at 506. Bolding mine.
How's "you can't tell if someone's trans" working out for you, TRAs?
Justice Alito's dismantling of the very idea that an internal identity can grant membership of an oppressed class is truly masterful.
[1] This refers to a very approximate US equivalent to an English law being subject to judicial review because it breaches the Equality Act 2010.