Updated characterization conjecture for Turánable oriented graphs

Let $H$ be an oriented graph. For $r\geq 1$, let $F_r$ be the tournament on vertex set $[3]^r$ in which, for distinct $u=(u_1,\ldots,u_r)$ and $v=(v_1,\ldots,v_r)$, the edge is directed from $u$ to $v$ when $(u_i,v_i)\in\{(1,2),(2,3),(3,1)\}$ for the smallest index $i$ with $u_i eq v_i$. Equivalently, $F_1=C_3$, and $F_{r+1}$ consists of three vertex-disjoint copies of $F_r$ with all edges directed cyclically between the copies. Let $C_k^\ell$ denote the $\ell$-th power of a consistently oriented cycle on $k$ vertices.

Updated Turánability conjecture. The oriented graph $H$ is Turánable if and only if there are integers $r$ and $k$ such that

$$H\subset F_r$$

and

$$H\subset C_{2k+1}^k.$$

This conjecture is proposed as an updated characterization after the preceding characterization by containment in some $D_s$ was disproved. It is motivated by the tournament case, where containment in both an $F_r$ and a $C_{2k+1}^k$ is equivalent to containment in some $D_s$; the source gives no resolution of the updated conjecture.