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On what grounds do you think it's likely or unlikely that no such test exists? Is it intuition, because you assume they are wrong, or because you have some evidence that no such test exists? Is there any reason why people shouldn't try to come up with such a test?

To detect design we need some template to test design against: some criteria of a 'designed object'. This has to be objective - it has to include all the possible ways something could be designed, since we know nothing about the designer. But of course, the set that incluiudes every possible designer has the ability to create every possible design, since we can simply define designer x as the designer that would create design y. We therefore need some criteria to narrow the set of designers. Thus, the task of detecting design objectively is impossible.

People can waste their time any way they like. However, if they come up with some sort of shoddy fake, as Dembski has, and try to sell it to the rest of us as an objective way of detecting design, they can expect to be scathingly rebutted, as Dembski has been.

I don't think that's true. Looking at the fire pit, for example, it is just as useful to know what a natural fire leaves behind as it is to know what a man made fire leaves behind. If you only knew what a natural fire looked like, it would be possible to determine that a fire didn't look natural.

So tell me what a natural fire lools like.

As a semi-made up example, suppose natural fires are started by lightning and lightning very rarely strikes twice in the same place

That's an immediately bogus proposition. The probability of a second lightning strike at a given location is higher, not lower, than the probability of a first, because places that get struck by lightning generally have characteristics that make them more likely to be struck by lightning. But anyway...

If I come upon the remains of a fire that contains many layers of strata suggesting there were many different fires in the same place, I could simply assume that this spot was a random fluke of probability or I might start looking for another reason why there were repeated fires in the same spot and look to make sure that I'm interpreting my evidence correctly.

It's a space alien landing zone. They have a secret beacon under the earth, and when they land there, they char the earth.

You attribute it to humans, because you know humans make fires. You're not starting with a tabula rasa. You don't go looking for all the possible coincidental characteristics of the earth.

Let me give you a counterexample. About 10,000 years ago, most of the large land mammals in North America became extinct. Was this a mere coincidence that this coincided with the arrival of humans, or was it cause-and-effect? Tell me, knowing only those two facts, how you could tell.