On special equal sums $x_1^n+x_2^n +\dots + x_n^n = (x_n+1)^n$

Let all terms be positive. There are infinitely many solutions to,

$$a^2+b^2 = (b+1)^2$$

$$a^3+b^3+c^3 = (c+1)^3$$

like the well-known $3^2+4^2 = 5^2$ and $3^3+4^3+5^3 = 6^3$. However, this has versions for higher degrees.

For $4$th powers by Jaroslaw Wroblewski,

$$178^4 + 1345^4 + 10400^4 + 143020^4 = (143020 + 1)^4$$

and close ones like

$$\qquad 34^4+210^4+2630^4+14405^4=(14405+4)^4$$ $$5662^4+14735^4+18190^4+163600^4=(163600+9)^4$$

For $5$th powers by James Waldby,

$$389^5 + 437^5 + 645^5 + 690^5 + 2766^5 = (2766+1)^5$$ $$202^5 + 239^5 + 249^5 + 811^5 + 2898^5 = (2898+1)^5$$

For $7$th powers by Maurice Blondot,

$$91^7 + 148^7 + 158^7 + 255^7 + 258^7 + 309^7 + 625^7 = (625+1)^7$$

Note 1: Wroblewski's database for $(4,1,4)$ has 1009 solutions so maybe it was a game of numbers.

Note 2: However, there are only four known $(7,1,7)$, yet Blondot's solution just so happen to be of the relevant form.

Note 3: Waldby's database for $(5,1,5)$ is only up to sum $z<10000$, far from Wroblewski's $z < 220000$.

Question: So can we find more examples for 4th and 5th powers?