TITLE: Prominent examples of $q$-analogs without known cyclic sieving
QUESTION [8 upvotes]: The cyclic sieving phenomenon is nicely summarized in the following AMS Notices "What is...?" article: https://www.ams.org/notices/201402/rnoti-p169.pdf. 
In that article, Reiner, Stanton, and White explain some desiderata (conditions (i)-(vi)) for "very nice" $q$-analogs $X(q)$ of cardinalities $\#X$ of combinatorial sets.
Question: What are some "very nice" $q$-analogs (especially, those with simple product formulas) for which there is no known cyclic action which gives rise to a CSP?

REPLY [5 votes]: I think I can provide one example, from this paper, The Cone of Cyclic Sieving Phenomena by N. Amini and I.
We look at a principal specialization of Schur polynomials,
$$
s_{n \lambda}(1,q,\dotsc,q^k)
$$
for some partition $\lambda$, and integers $n$ and $k$.
Here, $n\lambda = (n\lambda_1,\dotsc,n\lambda_\ell)$.
This polynomial evaluates to non-negative integers at $n$th roots of unity, and I think one should be able to prove without effort that there should in principle be a cyclic group action which gives CSP for this family.
The results by Rhoades show that (a suitable chosen power of) promotion works as group action whenever $\lambda$ is a rectangle.
As a side note, I would be happy to compile a list of suspected instances of the cyclic sieving phenomenon on my web page.
At the moment, it lists (most?) published instances of CSP.
I also have a private list with suspected instances, but I prefer to save these for student projects, and a few others I am working on myself.