Mixed sums of squares and triangular numbers (II)

For an integer $x$ let $t_x$ denote the triangular number $x(x+1)\mathrm{/}2$. Following a recent work of Z. W. Sun, we show that every natural number can be written in any of the following forms with $x,y,z\in \mathbb{Z}$: $$x^2+3y^2+t_z,x^2+3t_y+t_z,x^2+6t_y+t_z,3x^2+2t_y+t_z,4x^2+2t_y+t_z\mathrm{.}$$ This confirms a conjecture of Sun.