The four-antipalindrome conjecture

An antipalindrome is a natural number whose canonical base-$2$ expansion equals the complement of its reversal. The claim concerns all even natural numbers.

Four-antipalindrome conjecture. Every even natural number is the sum of at most $4$ antipalindromes.

The source says that the conditional bound of $13$ is probably far from optimal and that numerical evidence supports this stronger bound. No proof is given, so the conjecture remains open.