Maximality conjecture for the universal renewal polynomial

Let $S$ be the random walk whose positive, finite, integer-valued increments have probability masses $p_1,\ldots,p_{k-1}$, let $u_n$ be its renewal masses, and let $m_k$ be the minimum of the relevant renewal masses. Let $Q_k$ be the universal polynomial satisfying $m_k\geq Q_k$, and let $\mathop{\mathtt{A}}_k$ and $\widehat{\mathop{\mathtt{A}}}_k$ be the polynomial classes defined in the paper.

Maximality conjecture. The polynomial $Q_k$ is a maximal element of $\mathop{\mathtt{A}}_k$ and is the largest element of $\widehat{\mathop{\mathtt{A}}}_k$ for every $k\geq 3$.

The conjecture concerns universal polynomial lower envelopes for the minimum of discrete renewal sequences. The paper proves that $Q_k$ is a lower bound and that $\mathop{\mathtt{A}}_k$ has no largest element when $k\geq 3$; the maximality and largest-element assertions remain conjectural.