Existence of a p-time generator whose range intersects every infinite P set

Let $g$ be a polynomial-time function that maps inputs of length $n$ to outputs of length $n+1$, and let $rng(g)$ denote its range. Proof search conjecture. There exists such a function $g$ whose range intersects every infinite ${\cal P}$ set; equivalently, the complement of $rng(g)$ is ${\cal P}$-immune. The source presents this as equivalent to the conjecture that there is a uniform generator search-hard for all proof systems, obtained by replacing ${\cal N}{\cal P}$ sets with ${\cal P}$ sets in the proof-search formulation. Its status is left open in the source.