The CM-lift conjecture for Hodge classes on abelian varieties

Let $w$ be a prime of $\mathbb{Q}^{\mathrm{al}}$ and let $A$ be an abelian variety over $\mathbb{Q}^{\mathrm{al}}$ with good reduction at $w$. Let $\gamma$ be a Hodge class on $A$. Two pairs $(A,\gamma)$ and $(A',\gamma')$ have isogenous specializations preserving the classes if there is an isogeny $A_{0}\to A'_{0}$ sending $\gamma_{0}$ to $\gamma'_{0}$. The CM-lift conjecture. There exist a CM abelian variety $A'$ over $\mathbb{Q}^{\mathrm{al}}$ and a Hodge class $\gamma'$ on $A'$ such that $(A,\gamma)_{0}$ and $(A',\gamma')_{0}$ have isogenous specializations preserving the classes. This refines the fact that abelian varieties over the residue field lift up to isogeny to CM abelian varieties; the source gives approaches and partial cases but does not state a general proof.