$(x,y,z)=(2^{\frac{a-2}{2}},5^{\frac{a}{2}},5^a-2^{a-1}+2\cdot5^{\frac{a}{2}}\cdot 2^{\frac{a-2}{2}})$ works for $a$ large even number. Note that the expression can be factorized as $(2x^2+y^2-2xy+z)(2x^2+y^2+2xy-z)$, we want this product to be a power of $10$, a power of $10$ plus one, or twice a power of $10$, for the latter case we can get a solution by letting $2x^2+y^2=5^a+2^{a-1}$ and $2xy-z=2^{a-1}-5^a$.

Do you know when the results are being released??

Interesting problem!

$(x, y, z) \in \{(x, x, 5x^2)|x \in N\}$ yields $P(x, y, z) = 4x^4 + y^4 - z^2 + 4xyz = 0$

Another construction: Take $(x,y,z)=(5^n*2^{n-1}, 10^n, 5^{2n} *2^{2n-1})$ for every natural $n$. It is easy to see that $$4x^4+y^4-z^2+4xyz=2*10^{4n}$$a number whose sum of digits is exactly $2$.