Gray–Wyner conjecture for the doubly symmetric binary source

Let $(X,Y)$ be a doubly symmetric binary source ${\mathrm{DSBS}}(p)$ with $p\in(0,1/2)$. Let $R(D_{1},D_{2})$ denote its rate-distortion function under Hamming distortion, let $h$ be the binary entropy function, and let $R_{0}(R_{1},R_{2})$ be the minimum common rate in the Gray–Wyner coding system when the private rates are $(R_{1},R_{2})$. Gray–Wyner conjecture. For every $(R_{1},R_{2})\in[0,1]^{2}$,

$$R_{0}(R_{1},R_{2})=R(h^{-1}(R_{1}),h^{-1}(R_{2})).$$

Equivalently, the layered coding scheme that first uses an optimal point-to-point lossy compression code with distortions $(D_{1},D_{2})$ and then losslessly compresses the residuals at rates $h(D_{1})$ and $h(D_{2})$ should be optimal for the Gray–Wyner system for the DSBS. The conjecture has been open since 1974, with the source stating that no progress had been made on it at the time of publication.