Sun's refined 1-3-5 conjecture with a square or three-times-square coordinate

Let $x,y,z,t\in\mathbb{N}_0$. A representation of a positive integer as four squares is an expression

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

Sun's refined 1-3-5 conjecture. Any positive integer can be represented in this way with $x+3y+5z$ a square, and with at least one of the following properties: $x$ is three times a square, $y$ is a square, or $z$ is a square. The paper introduces this as a conjecture used in its computational work, but the supplied text does not state a resolution of this refinement.