Four positive binary squares conjecture

A positive binary square is the square of a positive integer whose binary expansion is used in the representation. The claim concerns representations of natural numbers as sums of exactly four such squares.

Four positive binary squares conjecture. Every natural number $N>1772$ is the sum of exactly four positive binary squares. The $112$ exceptions are

$$\begin{gathered} 0,1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,20,21,22,23,25,27,28,29,30,32,34,35,\\ 37,39,41,42,44,46,47,49,51,53,56,58,62,65,67,74,83,88,95,100,104,107,109,113,116,\\ 122,125,131,134,140,143,148,149,155,158,160,161,167,170,173,175,182,184,368,385,\\ 402,407,419,424,436,441,458,475,492,509,526,543,552,560,569,587,599,608,613,\\ 620,625,638,647,653,671,686,698,713,1508,1541,1574,1607,1640,1673,1706,1739,1772. \end{gathered}$$

The statement is presented as a likely consequence of extending the paper's method, and the supplied context gives no proof or resolution.