The initial-means conjecture for discrete geometric distributions

Let $k\geq 2$, let $(p_1,p_2,\ldots,p_{k-1})$ be a permutation of $\{1/2,1/2^2,\ldots,1/2^{k-1}\}$, and define the probability measure

$$P=\sum_{j=1}^{k-1}p_j\delta_j+\sum_{j=k}^{\infty}\frac{1}{2^j}\delta_j$$

on $\mathbb{R}$, supported on $\mathbb{N}$. Let $\{a_1,a_2,\ldots,a_n\}$ be an optimal set of $n$-means with $n\geq k+2$ and $a_1<a_2<\cdots<a_n$. Initial-means conjecture. The first $n-3$ means are the corresponding natural numbers: $a_1=1,a_2=2,\ldots,a_{n-3}=n-3$. This describes the structure of optimal quantizers for the discrete distribution and is used as the initial step toward determining their remaining means and quantization error; the supplied text does not indicate whether this assertion has been proved or remains open.