Majorization characterization conjecture for unimodal jump distributions

Let $p$ and $q$ be symmetric unimodal weight functions, meaning that $i\mapsto p(i)$ and $i\mapsto q(i)$ are symmetric and decreasing for $i\geq0$. Let $p\preceq_Mq$ denote majorization, and let $Z_t^p$ and $Z_t^q$ be the corresponding partition functions. Majorization characterization conjecture. Assuming that $p$ and $q$ are symmetric and unimodal,

$$p\preceq_Mq\quad\Longleftrightarrow\quad Z_t^q\preceq_{cv}Z_t^p$$

for all probability measures $\mathbb P$ satisfying the paper's spatial and integrability assumptions. Since majorization alone is not sufficient without the symmetry and unimodality restriction, this conjecture proposes that the restriction makes majorization both necessary and sufficient; it remains open.