Arithmetic autocorrelation value distribution conjecture for binary m-sequences

Let $\mathcal{M}$ be a binary $m$-sequence of period $2^n-1$, where $n\in\mathbb{N}$. For an integer shift $\tau$ with $1\leq\tau<2^n-1$, let $\mathcal{A}^{A}_{\mathcal{M}}(\tau)$ denote the arithmetic autocorrelation of $\mathcal{M}$. Arithmetic autocorrelation conjecture. (1) The arithmetic autocorrelation satisfies

$$\mathcal{A}^{A}_{\mathcal{M}}(\tau)\in\{\pm(2^k-1):1\leq k<n\}.$$

(2) For each $k$ with $1\leq k<n$, there are $2^{n-k}$ shifts $\tau$ satisfying $1\leq\tau<2^n-1$ and

$$\left|\mathcal{A}^{A}_{\mathcal{M}}(\tau)\right|=2^k-1.$$

The conjecture is motivated by numerical experiments for binary $m$-sequences, whose observed arithmetic autocorrelation values agree with the stated set and distribution. Its resolution would describe the complete value distribution of arithmetic autocorrelation for these sequences; the source leaves related questions for other generalized cyclotomic generators open.