Can we write each positive integer as $w^2+x^2(1+2y^2+2z^2)$ with $w,x,y,z\in\mathbb Z$ and $x\not=0$?

Lagrange's four-square theorem states that each nonnegative integer is the sum of four squares. Here I ask the following question concerning a refinement of Lagrange's four-square theorem.

QUESTION: Can we write each positive integer as $w^2+x^2(1+2y^2+2z^2)$ with $w,x,y,z\in\mathbb Z$ and $x\not=0$?

I conjecture that it has a positive answer $($see http://oeis.org/A275738 for related data, e.g., $2033=33^2+4^2(1+2\times 2^2+2\times 5^2))$, but I'm unable to show this. As $$2y^2+2z^2=(y+z)^2+(y-z)^2,$$ a positive answer to the question implies Lagrange's four-square theorem. I have used the theory of ternary quadratic forms to obtain that any positive integer can be written as $x^2+4^k(1+y^2+4z^2)$ with $k,x,y,z$ nonnegative integers (see my paper http://dx.doi.org/10.1016/j.jnt.2016.11.008).

Similarly, I also conjecture that any positive integer can be written as $x^2+4^k(1+y^2+5z^2)$ (or $5x^2+4^k(1+y^2+z^2)$) with $k,x,y,z$ nonnegative integers (cf. http://oeis.org/A275675 and http://oeis.org/A275676).

Maybe some experts at quadratic forms could answer my question. Any comments are welcome!