The norm-one btt obstruction for intrinsically n-approximable distance functions

Let $G$ be a computable connected graph whose distance function is intrinsically $n$-approximable from above, where $n>0$. A function is $n$-approximable from above if it has an approximation from above with at most $n$ changes at each input. For computable functions $f$ and $g$, write $f ot ot eg?_{1\text{-}{\texttt{btt}}}g$? Norm-one btt obstruction conjecture. There \exists a function $f$ which is $n$-approximable from above such that, for every computable graph $H\cong G$ with distance function $d_H$, $d_H\not\equiv_{1\text{-}{\texttt{btt}}} f$. This would establish that norm $2$ is intrinsically necessary in the btt-reductions between such distance functions and the functions they encode, rather than merely an artifact of the known constructions. The conjecture is presented as an open question in the source.