6 problems

Spring-free interval-order conjecture. Every spring-free, $\mathbf{4}+\mathbf{1}$-free interval order is 2-count.

Extension conjecture. The equality between $F_{\text{1}}(x,y)$ and $F_{\text{3}}(x,y)$ extends to those complex values of $p$ and $q$ for which both sides are defined.

Let $P$ be an interval order, let $h(P)$ denote its height, and let $G_P$ be its cover graph. Felsner–Trotter's conjecture. If … then … This gives a conjectured sharp relationship…

Let $r_n$ be the number of self-dual interval orders of size $n$, and let $q_n$ be the number of primitive self-dual interval orders of size $n$. Size asymptotic conjecture. … with…

OEIS A158691 coefficient conjecture. According to the notes in the OEIS entry, the coefficients of this power series are equal to $q_m$ for $m\ge 1$.

Let $F(x,y)$ be the generating function of interval orders, where $x$ counts the size and $y$ counts the number of minimal elements. Kitaev–Remmel's conjecture. They conjectured th…