Finding more solutions to seventh powers $(7,4,4)$ below a bound?

I. Manifolds

A non-singular homogeneous polynomial of degree $n+2$ with $n+2$ variables is a compact Calabi-Yau manifold, some of which important in string theory. For $n=2,3,5$ dimensions, we have,

$$x_1^4+x_2^4+x_3^4+x_4^4 = 0$$ $$x_1^5+x_2^5+x_3^5+x_4^5+x_5^5 = 0$$ $$x_1^7+x_2^7+x_3^7+x_4^7+x_5^7+x_6^7+x_7^7 = 0$$

The first one, in the form $x_1^4+x_2^4+x_3^4 = x_4^4\,$ has infinitely many primitive integer solutions. The second or the Fermat quintic threefold discussed here, has four known solutions. The third has near-misses,

$$2787^7 + 1951^7 + 1031^7 + 130^7 - 2816^7 - 1348^7 - 1146^7 = 1$$

where the sum almost vanishes. Question: Can we find more integers $N$ that can be expressed as,

$$N = x_1^7+x_2^7+x_3^7+x_4^7$$

in two ways for non-negative integers $0\le x_i < B$ for some bound $B$, or a $(7,4,4)$ in the notation by Meyrignac et al? (And with a higher search radius, the hope is one term $x=0$, hence satisfying the last manifold above.)

II. Data

The table is a summary of 38 known primitive solutions found mostly between 1999-2002 with "slow" computers. From the number of solutions per range, one may conclude it was NOT an exhaustive search.

\begin{array}{|c|c|} \hline \text{Range of largest x} & \text{# of solutions}\\ \hline \,\, 101-1000 & \;9 \\ \hline 1001-2000 & 10\\ \hline 2001-3000 & \;\color{red}4\\ \hline 3001-4000 & 15\\ \hline \text{Total} & 38\\ \hline \end{array}

More than 30 were found by Nuutti Kuosa while the rest was by Ekl, Lau, Choudhry, and Gascoigne. We can use the bound $x_i<4000$ for convenience.

Note: In the data below, the largest term $x$ is in the leftmost position.

1st range

149^7 + 123^7 + 14^7  + 10^7 - 146^7 - 129^7 - 90^7 - 15^7
194^7 + 150^7 + 105^7 + 23^7 - 192^7 - 152^7 - 132^7 - 38^7
354^7 + 112^7 + 52^7  + 19^7 - 343^7 - 281^7 - 46^7  - 35^7
554^7 + 412^7 + 365^7 + 146^7 - 536^7 - 439^7 - 437^7 - 65^7
591^7 + 315^7 + 299^7 + 97^7  - 553^7 - 517^7 - 208^7 - 66^7
639^7 + 328^7 + 281^7 + 175^7 - 581^7 - 555^7 - 471^7 - 320^7
698^7 + 556^7 + 443^7 + 184^7 - 673^7 - 625^7 - 353^7 - 230^7
711^7 + 688^7 + 332^7 + 313^7 - 698^7 - 668^7 - 592^7 - 86^7
984^7 + 471^7 + 353^7 + 221^7 - 909^7 - 861^7 - 619^7 - 354^7

2nd range

1045^7 + 315^7 + 91^7  + 73^7  - 1002^7 - 791^7 - 765^7 - 184^7
1105^7 + 666^7 + 431^7 + 314^7 - 1098^7 - 752^7 - 536^7 - 130^7
1307^7 + 857^7 + 618^7 + 400^7 - 1184^7 - 1133^7 - 1030^7 - 423^7
1341^7 + 1168^7 + 626^7  + 323^7 - 1298^7 - 1243^7 - 496^7 - 421^7
1403^7 + 797^7  + 606^7  + 529^7 - 1384^7 - 1012^7 - 744^7 - 489^7
1456^7 + 1095^7 + 1040^7 + 487^7 - 1395^7 - 1315^7 - 383^7 - 313^7 
1478^7 + 423^7  + 327^7  + 103^7 - 1412^7 - 1198^7 - 946^7 - 77^7
1555^7 + 1112^7 + 902^7  + 344^7 - 1535^7 - 1237^7 - 662^7 - 479^7
1901^7 + 849^7  + 610^7  + 202^7 - 1763^7 - 1673^7 - 869^7 - 685^7
1924^7 + 1446^7 + 1430^7 + 1095^7 - 1862^7 - 1735^7 - 468^7 - 360^7 

Note: It seems there's a gap after $x = 1555$.

3rd range (?)

2352^7 + 1545^7 + 1498^7 + 1238^7 - 2199^7 - 2119^7 - 706^7 - 307^7
2816^7 + 1348^7 + 1146^7 + 1^7   - 2787^7 - 1951^7 - 1031^7 - 130^7
2910^7 + 2522^7 + 2089^7 + 526^7 - 2857^7 - 2690^7 - 1480^7 - 1020^7
2982^7 + 2268^7 + 2252^7 + 1647^7 - 2946^7 - 2385^7 - 2325^7 - 65^7

Note: Obviously incomplete. But a term with $x=1$ appears. Maybe also a $x=0$?

4th range

3018^7 + 2183^7 + 1600^7 + 274^7 - 2816^7 - 2703^7 - 1831^7 - 1489^7
3102^7 + 1485^7 + 721^7  + 695^7 - 3053^7 - 2145^7 - 1927^7 - 264^7
3178^7 + 1361^7 + 747^7  + 648^7 - 3008^7 - 2607^7 - 2093^7 - 1796^7
3121^7 + 1960^7 + 403^7  + 311^7 - 2966^7 - 2666^7 - 1522^7 - 699^7
3189^7 + 1823^7 + 1140^7 + 1^7   - 3188^7 - 1859^7 - 621^7 - 485^7
3225^7 + 2804^7 + 1743^7 + 661^7 - 3133^7 - 2978^7 - 340^7 - 92^7
3245^7 + 1268^7 + 758^7  + 354^7 - 2951^7 - 2928^7 - 833^7 - 677^7
3270^7 + 2687^7 + 1482^7 + 281^7 - 3242^7 - 2714^7 - 2093^7 - 427^7
3271^7 + 2790^7 + 1528^7 + 914^7 - 3135^7 - 3031^7 - 1486^7 - 305^7
3476^7 + 3004^7 + 2435^7 + 1741^7 - 3328^7 - 3280^7 - 2111^7 - 1937^7
3586^7 + 2725^7 + 1324^7 + 684^7  - 3568^7 - 2810^7 - 1189^7 - 752^7
3790^7 + 2008^7 + 1611^7 + 1133^7 - 3538^7 - 3321^7 - 767^7 - 34^7
3798^7 + 3438^7 + 2640^7 + 567^7 - 3754^7 - 3330^7 - 3127^7 - 400^7
3891^7 + 2768^7 + 1012^7 + 56^7  - 3593^7 - 3223^7 - 3196^7 - 403^7
3930^7 + 1235^7 + 1126^7 + 18^7  - 3923^7 - 2106^7 - 1138^7 - 612^7

Note: Another $x=1$. And it seems also incomplete since the density after $x = 3300$ drops off.

III. Question

The 3rd range (maybe all) is incomplete. To repeat the question, after 20+ years, can we now find new and more solutions below a reasonable bound $B$, maybe just $x_i<3000$ to fill in the 3rd range, especially if one has a term $x = 0$?

P.S. About $1/4$ or $25\text{%}$ of the solutions above are multi-grade and valid for exponents $k=(1,7)$, and why that is so is a good question in itself. In fact, three are valid for $k=(1,3,7)$ like,

$$698^k + 556^k + 443^k + 184^k = 673^k + 625^k + 353^k + 230^k$$

Choudhry and Wroblewski found a multi-variable cubic which yields a $k=(1,3,7)$, but the terms get large and most go beyond the bound $x_i>4000$ in this post.