Conjecture on the impossibility of universally ultra-tight oracle-use bounds

Let $X$ and $Y$ be infinite binary sequences, and let an oracle computation of $X$ by $Y$ have an oracle-use function measuring how many bits of $Y$ are queried to compute the first $n$ bits of $X$. Write $X\upharpoonright_n$ for the length-$n$ prefix of $X$, and let $K(X\upharpoonright_n)$ denote its prefix-free Kolmogorov complexity. Oracle-use conjecture. There exists $X$ such that, for every $Y$ computing $X$, the oracle-use in the computation is not bounded above by the function $n\mapsto K(X\upharpoonright_n)$. This asserts that the stronger bound suggested by the general upper bound $K(X\upharpoonright_n)+\log n$ is not achievable for every sequence. Bounds of this form are connected with online coding methods and the complexity of initial segments; the cited special case of left-c.e. reals shows that a related bound can be achieved in some cases, while the general impossibility claim is presented here as a conjecture.