Let $a>b>0$, and let $T=\{a,b\}^n$ be the set of all $n$-tuples each entry of which is $a$ or $b$. Let $X\subseteq\{0,1\}^n$ with $|X|>1$, and let $f:T\rightarrow X$ be a function. For each $\textbf{t}\in T$, the score $s(f,\textbf{t})$ is defined as $\max_{x\in X}\dfrac{f(\textbf{t})\cdot \textbf{t}}{\textbf{x}\cdot\textbf{t}}$. The total score $s(f)$ is defined as $\min_{t\in T}s(f,t)$.
Suppose Ann plays a game with Bob. Ann can choose $a,b,X,f$ (possibly as functions of $n$). Bob knows $a,b$, doesn't know $X$, and can only query $f$ from Ann as will be described. Bob's task is to "design" a function $g:T\rightarrow X$. Ann will present Bob with $\textbf{t}_1,\textbf{t}_2,\ldots,\textbf{t}_{2^n}\in T$ (one at a time, in any order she wants). When Bob gets a new $\textbf{t}_i$, Bob can ask Ann for polynomially many $f(\textbf{t})$'s and then tell Ann his choice $g(\textbf{t}_i)$. Bob has two constraints:
1) His choice $g(\textbf{t}_i)$ must be among the $f(\textbf{t})$'s that Ann has answered him previously in the game.
2) For $\textbf{t}$ and $\textbf{t}'$ that differ in only one coordinate $i$, with $\textbf{t}$ having $a$ and $\textbf{t}'$ having $b$, Bob cannot choose so that $g(\textbf{t})$ has $0$ and $g(\textbf{t}')$ has $1$ in coordinate $i$.
Does there exist a constant $c$, independent of $n$, such that Bob can always design $g$ for which $s(g)\geq c\cdot s(f)$, no matter what Ann does?
(Note that without the second condition, the question is trivial. Indeed, Bob can just choose $g(\textbf{t}_i)=f(\textbf{t}_i)$ and get $s(g)=s(f)$.)