Degree-square Turán problem for two self-converse tournaments

Abstract: For a fixed digraph $F$, let $\operatorname{ex}<em>2<sup>+(n,F)$ be the maximum of $\sum</em>{v\in V(D)}d_D<sup>+(v)<sup>2$ over all $n$-vertex $F$-free digraphs. Ai et al. [arXiv:2606.03520, 2026] asked for which self-converse tournament $F$ one can determine $\operatorname{ex}_2<sup>+(n,F)$. Let $TT_r$ denote a transitive tournament on $r$ vertices and $\text{RT}_5$ denote a regular tournament on 5 vertices. Note that both $TT_r$ and $\text{RT}_5$ are self-converse. Iľkovič [arXiv:2609.05042, 2026] determined $\operatorname{ex}_2<sup>+(n,F)$ for $F=TT_4$. In this paper, we determine $\operatorname{ex}_2<sup>+(n,F)$ for $F$ being $TT_r$ or $\text{RT}_5$, and the latter confirms a conjecture due to Iľkovič.