Error-resistant protocols for generalized Ulam's game

Let $X_1,\ldots,X_k$ and $Y$ be sets, let $f\colon X_1\times\cdots\times X_k\to Y$ be a function, and let ${\cal G}\left<f,q,l\right>$ denote the generalized Ulam game in which Paul asks at most $q$ coordinate-membership questions and Carole may lie at most $l$ times. A winning strategy for Paul identifies the unique possible value of $f$. Generalized Ulam game conjecture. There exists a constant $\varepsilon_g>0$ such that, for every function $f$ for which Paul has a winning strategy in ${\cal G}\left<f,n,0\right>$, if $l<\varepsilon_gq$, then Paul has a winning strategy in ${\cal G}\left<f,q,l\right>$. Moreover, $q\leq C_g n$, where $C_g$ is a constant depending only on $l/q$. The conjecture extends the known linear-question behavior of standard Ulam's game with a bounded fraction of lies to the generalized setting; its resolution would provide error-resistant protocols for arbitrary functions admitting a lie-free strategy.