Euclidean cone curvature-dimension conjecture

Let $X$ be an essentially non-branching metric-measure space, and let $C(X)$ denote its Euclidean $N$-cone. Euclidean cone curvature-dimension conjecture. The Euclidean $N$-cone $C(X)$ satisfies $\mathsf{CD}(0,N+1)$ if and only if $X$ satisfies $\mathsf{CD}(N-1,N)$. This is the interval-base special case of the preceding warped-product conjecture and extends the corresponding established statement with the stronger $\mathsf{RCD}$ condition; the source gives no resolution of the $\mathsf{CD}$ version.