Conjecture on decomposing CD-affine needles under a zero-mean constraint

Let $f535$ be a probability measure on $f535$ that is a $CD(\kappa,N+1)$-needle, where $N\in(0,+\infty]$ and $\kappa\geq0$. Let $\varphi:\mathbb R\to\mathbb R$ be continuous and $\mu$-integrable, with

$$\int_{\mathbb R}\varphi\,d\mu=0.$$

A $CD(\kappa,N+1)$-affine needle is a one-dimensional $CD(\kappa,N+1)$-needle satisfying the affine equality case described in the source. CD-affine needle decomposition conjecture. There should exist a probability measure $\nu$ on a set $\Omega$ and probability measures $\{\mu_\alpha\}_{\alpha\in\Omega}$ on $\mathbb R$ such that $\mu(A)=\int_\Omega\mu_\alpha(A)\,d\nu(\alpha)$ for every Lebesgue-measurable $A\subseteq\mathbb R$, and, for $\nu$-almost every $\alpha$, $\mu_\alpha$ is either supported on a singleton or is a $CD(\kappa,N+1)$-affine needle satisfying $\int_{\mathbb R}\varphi\,d\mu_\alpha=0$. Such a decomposition would refine one-dimensional $CD(\kappa,N)$ localization into explicit affine needles; the source presents it as a proposed extension, and no resolution is supplied.