Existence of regular open and closed XOR-magic graphs of power-of-two order

Let $n$ be a positive integer, and let $d$ be a degree. An open XOR-magic graph and a closed XOR-magic graph are graphs admitting, respectively, an open or closed XOR-magic labeling. The graphs below are required to have order $2^n$.

For each degree

$$d\in\{5,7,\ldots,2^n-7,2^n-5\},$$

there exists a $d$-regular open XOR-magic graph of order $2^n$. For each degree

$$d\in\{4,6,\ldots,2^n-6,2^n-4\},$$

there exists a $d$-regular closed XOR-magic graph of order $2^n$.

Existence conjecture. For every permitted degree $d$ in the two displayed sets, the corresponding regular open or closed XOR-magic graph exists. This would complete the existence picture apart from the degrees excluded by the previously stated nonexistence result and extend the constructions known for order $16$ and order $8$ to every order that is a power of two.