Upper density of the Ulam sequence

The Ulam sequence is the unique sequence of positive integers defined as follows. Let a unique Ulam sum for a sequence $a : \mathbb{N} \to \mathbb{N}$ at stage $n$ be an integer $m$ that can be written uniquely as $a(i) + a(j)$ with $i < j < n$. The Ulam sequence is the sequence $a$ with $a(0) = 1$, $a(1) = 2$, and for each $n \geq 2$, the value $a(n)$ is the least integer greater than $a(n-1)$ that is a unique Ulam sum at stage $n$. For a subset $S \subseteq \mathbb{N}$, the upper density is defined as $\limsup_{N \to \infty} \frac{|S \cap \{1, 2, \ldots, N\}|}{N}$. Determine whether the upper density of the range of the Ulam sequence equals $0$.