New solutions for the equation $x_1^8+x_2^8+x_3^8+x_4^8=y_1^8+y_2^8+y_3^8+y_4^8$ (8,4,4)

for the diophantine $x_1^8+x_2^8+x_3^8+x_4^8=y_1^8+y_2^8+y_3^8+y_4^8 $

Back in 2006, Nuutti Kuosa discovered the following non-trivial integer solution $$1953^8+2012^8+3113^8+861^8=1128^8+2767^8+2557^8+2823^8$$ A system of equations was used to speed up the search. $$x_1-x_2-x_3+2x_4=-y_1+y_2+y_3-2y_4$$ $$2x_1-x_2-x_3=y_1+y_2-2y_3$$ There are too few open questions and studies on this equation on the Internet. A similar question was asked by TitoPiezasIII in this post, but there was no mention specifically of this equation.

Using brute force, it was possible to achieve heights of $x_i<500$, nothing was found within the range.

I managed to get about 90 examples of a simpler form of this diophantine equation by reducing 8 parameters to 7 $$x_1^{\color{blue}{4}}+x_2^{\color{blue}{4}}+x_3^8+x_4^8=x_5^8+x_6^8+x_7^{\color{blue}{4}}+x_8^{\color{blue}{4}}$$ $$a^8+b^8+(q+v)^4+(-q+v)^4=(s+v)^4+(-s+v)^4+c^8+d^8$$ $$27^8+58^8+15^8+267^4=11^8+20^8+53^8+2851^4$$ $$32^8+34^8+21^8+1007^4=27^8+8^8+7^8+1384^4$$ so far, I have not been able to obtain a shape with 7 eighth degrees in a small radius $x_i<250$

Parity and Modular Constraints

An analysis of $8\text{th}$ power residues modulo small integers drastically cuts down the search grid. By checking the allowable residues, we can establish strict structural filters:

• $\pmod{32}$: Since $x^8 \equiv 0$ or $1 \pmod{32}$.
• $\pmod{17}$: Residues are strictly limited to $\{0, 1, 16\}$.
• $\pmod{9}$: Residues are restricted to $\{0, 1, 4, 7\}$.

Are there other known nontrivial solutions to this equation $(8,4,4)$ ? Perhaps a good step for an answer would be to use auxiliary parametric families.

please correct the question if you see any inaccuracies in the wording, English is not my native language.