Having LLMs capable of generating text based on human training data obviously raises the bar for a text-only evaluation of "are you human?", but LLM output is still fairly easy to spot, and knowing what LLMs are capable of (sometimes superhuman), and not capable of, should make it fairly easy for a knowledgeable "turing test administrator" to determine if they are dealing with an LLM or not.
It would be a bit more difficult if you were dealing with an LLM agent tasked with faking a turing test as opposed to a naieve LLM just responding as usual, but even there the LLM will reveal itself by the things that it plain can't do.
LLM output might be harder to spot when it's mostly commands to drive the browser.
I often interact with the web all day and don't write any text a human could evaluate.
Easy to spot assuming the LLM is not prompted to use a deliberately deceptive response style rather than their "friendly helpful AI assistant" persona. And even then, I've had lots of people swear to me that an emoji laden not this--but that bundle of fluff looks totally like it could have been written by a human.
If you need a specialized skill set (deep knowledge of current LLM limitations) to distinguish between human and machine then I would say the machine passes the turing test.