Ext-quiver conjecture for descent algebras of type A

Let $n$ and $p$ be as in the descent algebra setting, let mathop{\Lambda}\nolimits^+_p(n) denote the indexing set of $p$-regular partitions, and let $Q_{n,p}$ be the Ext quiver of the descent algebra $\mathscr{D}_{n,{F}}$. For $\lambda\in\Lambda^+(n)$, write $(\lambda,p)=1$ if $p\nmid\lambda_i$ for every $i\in[1,\ell(\lambda)]$, and write $(\lambda,p)\neq1$ otherwise. Let $Q_{n,\infty}$ be the corresponding characteristic-zero Ext quiver, and let $\sim_p$ denote the equivalence relation used to identify the relevant labels. For vertices $v,w$ of a quiver $Q$, let $n^Q_{v,w}$ be the number of arrows from $v$ to $w$. Ext-quiver conjecture. For $\lambda,\mu\in\Lambda^+_p(n)$, the following hold: if $\lambda=\mu$, then $n^{Q_{n,p}}_{\lambda,\lambda}>0$ if and only if $(\lambda,p)\neq1$; if $\lambda\neq\mu$, there is an arrow $\lambda\to\mu$ in $Q_{n,p}$ if and only if there exist $\delta\sim_p\lambda$ and $\gamma\sim_p\mu$ such that there is an arrow $\delta\to\gamma$ in $Q_{n,\infty}$, and in this case $n^{Q_{n,p}}_{\lambda,\mu}=1$. This conjecture predicts the complete pattern of loops and arrows in the Ext quiver of the descent algebra in characteristic $p$ from the characteristic-zero Ext quiver and the $p$-equivalence classes.