The exceptional three-antipalindrome conjecture

An antipalindrome is a natural number whose canonical base-$2$ expansion equals the complement of its reversal. Let

$$E=\{8,18,28,130,134,138,148,158,176,318,530,538,548,576,644,1300,2170,2202,2212,2228,2230,2248,8706,8938,8948,34970,35082\}.$$

Exceptional three-antipalindrome conjecture. Every even natural number outside $E$ is the sum of at most $3$ antipalindromes.

The claim is presented as a numerical conjecture refining the preceding four-antipalindrome bound. The listed exceptions are the only exceptions asserted by the source, and no proof or resolution is provided.