Ultra-Euclidean functions are preserved by tilde transformation

Let $R$ be a Euclidean domain, let $f:R\setminus\{0\}\to\mathbb{Z}_0^+$ be an ultra-Euclidean function, and let $\widetilde{f}$ denote the associated transformed function. Preservation conjecture. If $f$ is ultra-Euclidean on $R$, so is $\widetilde{f}$. The supplied status evidence marks this conjecture as disproved; however, the surrounding text says that the authors were unable to prove or disprove it, so the resolution and the definition of $\widetilde{f}$ should be checked against the source.