$\overrightarrow{u_iu_{i'}}$ of $H$, orient all edges from $V_i$ to $V_{i'}$ in $T_1$. Orient the remaining edges of $T_1$ arbitrarily. As each map $\phi : V(H) \to V(T_1)$ with $\phi (u_i) \in V_i$ for all $i\in [h]$ is a labelled embedding, we see that $$\label{equation: T_1 count}
            N^*_{T_1}(H) \geq \big (h^{-h} - o(1) \big )n^h.$$ Now, starting from $T_1$, construct a sequence of tournaments $T_1, \ldots , T_{n}$ on vertex set $V(T_1)$, where for $i\geq 1$ each $T_{i+1}$ is obtained from $T_i$ by reorienting all edges to go from $v_i$ to $\{v_{i+1},\ldots , v_{n}\}$. Using that $h^{-h} > 2^{-\binom {h}{2}}$ for $h\geq 7$, by there is $\delta >0$ such that $N^*_{T_1}(H) > (2^{-\binom {h}{2}} + \delta ) n^h$ for large $n$. On the other hand $T_{n} = Tr_n$ and as $H$ is not transitive we have $N^*_{T_{n}}(H) = 0$. Since $N^*_{T_{i}}(H) = N^*_{T_{i+1}}(H) \pm hn^{h-1}$, by an intermediate value property $N^*_{T_{i}}(H) = (2^{-\binom {h}{2}} + o(1))n^h$ for some $i\in [n]$. However, $T_i$ is not quasirandom. Indeed as $\big (2^{-\binom {h}{2}} + o(1) \big )n^h = N^*_{T_{i}}(H) \geq (2^{-\binom {h}{2}} + \delta )n^h - ihn^{h-1}$ we have $i \geq \delta n/2h$. As $|d^+_{T_i}(v_j) - d^-_{T_i}(v_j)| = n-2j+1 \geq n/2$ for all $j\in 