The power-of-two values conjecture for the Conway-type sequence d(n)

Let $d(n)$ be the Conway-type integer sequence, and let $I_K$ denote its generation domains, with $I_{-1}=\{1,2\}$. Power-of-two values conjecture. The sequence satisfies

$$d(2^k)=0\quad\text{for }k\geq 1,$$

and

$$d(2^k+1)=-1\quad\text{for }k\geq 2.$$

The claim describes two exact values at the beginning of each power-of-two scale. The source reports that it has been numerically verified to high order of $k$, but no proof or resolution is provided there.