Vic Reiner- open conjectures, questions, problems

1. (with B. Rhoades)
   Rationality of the q-Ehrhart series and finite generation of the harmonic algebra.
   See Conjectures 1.1 and 5.5 in here.
   
2. (with A. Adams)
   Resolutions of simplicial poset face rings over their canonical parameters.
   See Conjecture 6.1 in here.
   
3. (with G. Tudose and the 2020 Polymath Jr. group)
   Hilbert series for subalgebra filtrations of cohomology of ordinary and Lagrangian Grassmannians.
   See Conjecture 1 in here, and Conjecture 1.4 in here.
   
4. (with J. Lewis)
   "Root system circuits are acutely disconnected".
   Is there a non case-by-case proof of Lemma 1.2 in here, which asserts this fact about finite real reflection groups:
   any set of roots forming a minimal linear dependence with positive coefficients has a disconnected graph of pairwise acuteness.
   
5. (with J. Lewis and D. Stanton)
   The invariants of GL_n(F_q) mod Frobenius powers conjecture.
   See Conjecture 1.1 in here.
6. (with D. Stanton and D. White)
   Are there insightful, non-brute force proofs of the cyclic sieving phenomena on ...
   • triangulations of a convex n-gon under n-fold rotation?
   • alternating sign matrices under 4-fold rotation?
   See the discussion at the end here.
   
7. (with J. Lewis and D. Stanton)
   A q-analogue of Hurwitz's formula.
   Is there a short proof of Theorem 1.1 in here,
   showing there are (qⁿ-1)^n-1 factorizations of a Singer cycle in GL_n(F_q) into n reflections,
   similar to proofs that n-cycles have n^n-2 factorizations into n-1 transpositions?
   
8. (with D. Armstrong and B. Rhoades)
   The parking space conjecture.
   See the Main Conjecture from Section 2.6 in here.
   
9. (with F. Saliola and V. Welker)
   The symmetrized k-subset-to-top shuffling operators commute.
   Is there a conceptual, non-inductive proof of Theorem 1.1 in here?
   
10. (with Y. Roichman)
    Diameter of the graph of reduced words for the longest element in type D_n.
    Is the lower bound given by the number of codimension two intersection subspaces (see Table 1.1 here) tight or not?
    
11. (with A. Miller)
    Characteristic polynomials det(UD-tI) of up-down maps in differential posets have a Smith form over Z[t].
    See Conjecture 1.1 in here.
    
12. The number of braid relations in reduced words for the longest element of S_n approaches Poisson with mean 1.
    See Conjecture 2 in here
    
13. (with A. Duval)
    The generalized Grone-Merris conjecture.
    See Conjecture 1.2 in here, which says that in a d-dimensional simplicial complex,
    the top dimensional Laplacian spectrum is majorized by the conjugate partition to the sequence of vertex degrees with respect to the d-simplices.