The exact Euclidean window of $\mathbb{Z}[\sqrt{14}]$

Let $R=\mathbb{Z}[\sqrt{14}]$, let $\pi =4+\sqrt{14}$, and let $\mathfrak{p}=(2,\sqrt{14})=\pi R$ be the prime ideal above $2$. Cavallar and Lemmermeyer studied the multiplicative weighted norms ${\phi }_t(x)=\mathrm{\mid }N(x)\mathrm{\mid }(t\mathrm{/}2{)}^{v_{\pi }(x)}$ and asked for which weights $t$ they are Euclidean functions on $R$. We show that the set of these weights, the Euclidean window of $\mathfrak{p}$, is $(\sqrt{5},T]$, where $T$ is the algebraic number $$T=\sqrt{\frac{8}{104051\sqrt{2}-55617\sqrt{7}}}=2.2468286690\dots =\sqrt{5}+0.0107606915\dots \mathrm{.}$$ Thus the window is nonempty but not open. We also obtain the integer-valued completely multiplicative Euclidean function $2^{v_{\pi }(x)}\mathrm{\mid }N(x){\mathrm{\mid }}^6$ on $R$.