The computational refinement of the 1-3-5 conjecture

Let $m\in\mathbb{N}$ be not divisible by $16$, and let $x,y,z,t\in\mathbb{N}_0$. A representation of $m$ as four squares is an expression

$$m=x^2+y^2+z^2+t^2.$$

The computational refinement of the 1-3-5 conjecture. Except for $m\in\{31,43,111,151,168,200,248,263,319,456,479,871,1752,1864,3544\}$, every such $m$ has a representation for which $x+3y+5z$ is a square and either $x$ is three times a square or $y$ is a square. Moreover, if $m>14\,485\,001\,848$, then there is such a representation with $x\in\{0,3\}$ or $y\in\{0,1\}$; disregarding multiples of $16$, exactly $\frac56$ of the numbers have a representation with $x=0$ or $y=0$, and the remaining $\frac16$ have one with $x=3$ or $y=1$. This is presented as a new conjecture based on the computational results, and the supplied text gives no resolution.