The short answer is: nobody knows. To me this problem looks even harder than the problem of limit densities or other open problems like whether the sequence is recurrent and mirror/reversal invariant.
The general opinion among the ones who studied the problem is that it's very unlikely that $d(1)$ doesn't exists (this is in fact another open question), and provided that it exists, your claim is actually much stronger than $d(1)=1/2$.
Some heuristics: let $T(w)$ be the operator associating to every word in $\{1,2\}^*$ the word $v$ starting with 1 and such that $w$ lists the lengths of the runs of $v$ (example: $T(1221)=122112$). If you believe that the parity of the lengths $\ell(T^k(w))$ "behaves like" a Bernoulli variable with probability $1/2$, then "it follows" that the sequence is recurrent and that $d(1)$ equals 1/2 (if it exists). But even assuming this strong heuristic viewpoint, your claim would still be far from obvious.