Dominating sets in subtournaments of the Paley tournament 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 $q=19$).
 A: This is a construction answering the updated question. The original problem looks much, much harder.
Let $N$ be very large in terms of $k$. Let $M=kN$ The vertex set of the tournament is $\mathbb{Z}/M\mathbb{Z}$. There are two kinds of edges. First, there are edges from $x$ to each of $x+1,x+2,x+3,\dotsc,x+N$. We call these circular edges. For each pair $x,y$ not connected by a circular edge, we choose the direction of the edge connecting $x$ and $y$ uniformly at random.
Note that the definition of the tournament is invariant under reversal of the arrows and reversal of the circular direction. So, I will speak of the ``domination number'' instead of in-domination and out-domination.
The domination number of every subtournament is at most $k$. Indeed, let $U$ be the vertex set of the subtournament. Let $x_1=\min U$, and then define inductively $x_{i+1}=\min U\setminus \{1,2,\dotsc,x_i+N\}$. It is clear then that $\{x_1,\dotsc,x_k\}$ is a dominating set (which could be of size smaller than $k$ if we run out of elements sooner).
I claim that the domination number of the tournament is at least $k$. Indeed, let $X=\{x_1,\dotsc,x_{k-1}\}$ be any $(k-1)$-element set of vertices, and let $R=X+\{1,2,\dotsc,N\}$ be the set of the vertices connected to them by circular outedges. Then $S=V(T)\setminus R$ is a set of size at least $N-k$. Let's compute the probability that $S$ is dominated by $X$ via non-circular (=random) edges. Well, that probability is simply $(1-2^{-(k-1)})^{|S|}<\exp\bigl(-2^{-(k-1)}(N-k)\bigr)$. If $N=4^k$, say, then this is much less than $\binom{M}{k-1}^{-1}$, and so the claim follows by the union bound over all $(k-1)$-element sets $X$.
