Conjecture on the maximum of Q for signature parameter r_2=1 in degree five

Let $Q(5,1,\{4\},(x,y,1,g))$ be the function obtained after reducing to $z=1$ and $g\neq\pm1$, and let $(x,y,z,g)$ denote its parameters. Maximum-value conjecture. The maximum of $Q(5,1,\{4\},(x,y,1,g))$ is $16.6965\ldots$ and is attained at

$$(x,y,z,g)=\left(\frac{1}{\sqrt{7}},-1,1,\frac{1}{2\sqrt{7}}\right).$$

This is a numerical conjecture about the optimization of the regulator-related function $Q$; the source provides no proof or resolution.