Let $T_1$ be an effectively axiomatized first order theory whose language have only finitely many primitive symbols.
Now let $L^{T_1}$ be the set of all formulas of the language of $T_1$.
Let $L^{T_1}_S$ be the set of all sentences of the language of $T_1$
Where a sentence is a well formed formula with no free variables.
Now Let $F$ be a definable meta-theoretic syntactical injection from $L^{T_1}_S$ to $L^{T_2}_S$, like for example in saying that $F$ is the bounding of all quantifiers in $\phi$ by the constant symbol $V$ by relation $\in$ for every $\phi$ in $L^{T_1}_S$, Formally:
$F:L^{T_1}_S \to L^{T_2}_S, F(\phi) = \phi^V$,
Clearly $F$ is injective.
Now if we have: $\forall \phi \ \epsilon \ L^{T_1}_S [(T_1 \vdash \phi) \implies (T_2 \vdash F(\phi))]$
Where "$\vdash$" is syntactical provability.
Then we'd say that $T_2$ extends $T_1$ through $F$.
If we have: $\forall \phi \ \epsilon \ L^{T_1}_S [(T_1 \vdash \phi) \iff (T_2 \vdash F(\phi))]$
We say that $T_2$ conservatively extends $T_1$ through $F$.
Question1: Is the following implication true?
If there exists a function $F$ such that $T_2$ conservatively extends $T_1$ through $F$; then for every function $G$ such that $T_2$ extends $T_1$ through $G$, we have $T_2$ conservatively extends $T_1$ through $G$.