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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?