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By $(f)$ you mean $(f_1,\ldots,f_k)$? If you have one element, it always forms a regular sequence. If you have more than one element, then even if they are all monomials, the minimal resolution is not very explicit. Explicit non-minimal resolutions always exist, e.g. bar-cobar (using the Com-Lie Koszul duality).
@Tri just for historical record (and for possible benefit of a random reader), I can say that Google search brought me to the page staff.fnwi.uva.nl/t.h.koornwinder/dutchmathjournals/… - where one can be discouraged by the fact that the third series does not seem to be available, but clicking at the link for the second series and searching through the website it brings works like a charm!
MathSciNet has some 250 citations, which narrows it down a bit. Plus, if one searches for certain keywords in titles and abstracts, that leaves just a handful of papers. For example, among the papers citing Stanley that have "Littlewood-Richardson" in the title, this may be relevant : doi.org/10.1093/imrn/rnq126
It would be helpful to add a link to the paper where this was 'famously conjectured'. Have you checked the papers citing it? That would be the best way to answer your question.
@BugsBunny good question! Apparently, not: the Koszul dual algebra is infinite-dimensional (the dual of the generator $x$ is not nilpotent) but the inverse of the Hilbert series of $A$ has zero as the coefficient of $t^4$, and these two things cannot happen simultaneously for a Koszul algebra.
@TC what I would do is compute the Gröbner basis over $\mathbb{Q}$ carefully: do not cancel any common factors, and do not divide by anything. Then, once you compute your Gröbner basis, you will have the finite many leading coefficients, and if $p$ is not a divisor of either of them, you are already good because the $\mathbb{Q}$-Gröbner basis with integer coefficients also works modulo such $p$, so you just have finitely many $p$ to check. Avoiding using Gröbner bases means that you'd like to use something very specific about a particular problem, and it is up to you to figure out what it is.
