Exceptional sets for four ternary quadratic forms

Let $\mathbb N$ denote the nonnegative integers and let $x,y,z\in\mathbb Z$. Exceptional-set conjecture. The following representation sets are exactly:

$$\left\{\frac{x(5x+1)}2+\frac{y(5y+1)}2+\frac{z(5z+1)}2:x,y,z\in\mathbb Z\right\}=\mathbb N\setminus\{1,10,19,94\},$$ $$\left\{x(5x+1)+\frac{y(5y+1)}2+\frac{3z(5z+1)}2:x,y,z\in\mathbb Z\right\}=\mathbb N\setminus\{1,5,32\},$$ $$\left\{x(5x+1)+2y(5y+1)+\frac{z(5z+1)}2:x,y,z\in\mathbb Z\right\}=\mathbb N\setminus\{1,5,70\},$$ $$\left\{x(5x+1)+\frac{y(5y+1)}2+\frac{5z(5z+1)}2:x,y,z\in\mathbb Z\right\}=\mathbb N\setminus\{1,5\}.$$

These claims give precise finite exceptional sets for four ternary quadratic representations. They are posed as conjectures after related corollaries, and the source supplies no resolution.