4 problems

Asymptotically Catalan conjecture. There is no tile counting function $f\in\mathcal F$ that is asymptotically Catalan.

Smaller-height extension conjecture. Theorem also extends to tiles with smaller heights.

Catalan non-representability conjecture. The Catalan numbers $C_n$ are not a tile counting function.

Half-integer exponent conjecture. Every tile counting function has, on each residue class modulo some $m$, an asymptotic of the displayed form with a half-integer exponent $\alpha$…