For a tournament $T$, let $\mathrm{dom}(T)$ be the order of a smallest dominating set in $T$. Let $q$ be a prime power congruent to 3 mod 4 and let $T_q$ be the Paley tournament on $q$ vertices.

Is it true that for every subtournament $T$ of $T_q$ we have $\mathrm{dom}(T)\leq \mathrm{dom}(T_q)$?

There is quite a bit of research regarding bounds on $\mathrm{dom}(T_q)$, but I haven't found anything which is along the lines of my question -- where the actual value of $\mathrm{dom}(T_q)$ is irrelevant.

A good overview of what is known about the order of the smallest dominating sets in Paley tournaments can be found here Large dominating sets in tournaments and here https://www.win.tue.nl/~aeb/graphs/Paley.html

**Update:**

As suggested in the comments, let me state a more general version of my question. First, let's say that $S$ is an out-dominating set in $T$ if for all $v\in V(T)\setminus S$, there exists $u\in S$ such that $(u,v)\in E(T)$ and $S$ is an in-dominating set in $T$ if for all $v\in V(T)\setminus S$, there exists $u\in S$ such that $(v,u)\in E(T)$. Now define $\mathrm{dom}^+(T)$ to be the order of a smallest out-dominating set (compared to what I wrote above, $\mathrm{dom}^+$ and $\mathrm{dom}$ mean the same thing) and $\mathrm{dom}^-(T)$ to be the order of a smallest in-dominating set.

Is it true that for infinitely many $k\geq 2$, there exists a tournament $T$ such that $\mathrm{dom}^-(T)=k$ and for all $T'\subseteq T$, $\mathrm{dom}^+(T')\leq k$.

This would be implied by my more specific first question because if $T_q$ is a Paley tournament, then $\mathrm{dom}^+(T_q)=\mathrm{dom}^-(T_q)$ and for all $m\geq 2$, there exists $q$ such that $\mathrm{dom}^-(T_q)\geq m$.

A result of E. Szekeres and G. Szekeres says that if $|V(T)|<(k+2)2^{k-1}-1$, then $\mathrm{dom}^+(T)\leq k$. Since $\mathrm{dom}^-(T_7)=3$ and $\mathrm{dom}^-(T_{19})=4$, the result of Szekeres and Szekeres implies a positive answer to the general question for $k=2$ and $k=3$ (and the original question for $q=7$ and $q=19$).