OEIS sequence A297895 describes
Numbers that can be partitioned into squares of distinct integers whose reciprocals sum to 1.
1, 49, 200, 338, 418, 445, 486, 489, 530, 569, 609, 610, 653, 770, 775, 804, 845, 855, 898, 899, 939, 978, 1005, 1019, 1049, 1065, 1085, 1090, 1134, 1194, 1207, 1213, 1214, 1254, 1281, 1308, 1356, 1374, 1379, 1382, 1415, 1434, 1442, 1457, 1458, 1459, 1475, 1499, 1502, 1522, 1543, 1566, 1570, 1582
For example, $$ 49 = 2^2 + 3^2 + 6^2 \text{ and } \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 1;\\ 200 = 2^2 + 4^2 + 6^2 + 12^2 \text{ and } \frac{1}{2} + \frac{1}{4} + \frac{1}{6} + \frac{1}{12} = 1. $$
The author claims that
All integers $\geq 8543$ belong to this sequence.
What is a proof (or even a heuristic) that explains why this is the case?
As promised, here is my preprint with a proof that all integers $\geq 8543$ belong to A297895:
On partitions into squares of distinct integers whose reciprocals sum to 1
Questions and comments are welcome.