Density conjecture for sums of three binary squares

Let $S_3$ be the set of natural numbers that are the sum of three binary squares, and let $\alpha_3$ denote its lower asymptotic density.

Density conjecture. The lower asymptotic density satisfies

$$\alpha_3<0.9.$$

This conjecture is motivated by numerical evidence concerning representations as sums of binary squares. The supplied context does not indicate whether the conjecture has been resolved.