Uniform relativization conjecture for computable randomness under layerwise maps

Assume that $\mathbb{X}$ and $\mathbb{Y}$ are computable metric spaces, that $\{\mu_a\}_{a\in\mathbb{N}^{\mathbb{N}}}$ is a uniformly computable family of probability measures on $\mathbb{X}$, and that $\{T_a\}_{a\in\mathbb{N}^{\mathbb{N}}}$ is a family of layerwise maps $T_a\colon(\mathbb{X},\mu_a)\rightarrow\mathbb{Y}$, with $T_a$ Schnorr layerwise computable uniformly in $a$. For $a_0\in\mathbb{N}^{\mathbb{N}}$, say that $x\in\mathbb{X}$ is $\mu_{a_0}$-computably random uniformly relativized to $a_0$ when it passes every uniformly relativized computable randomness test as defined in the setup. Uniform relativization conjecture. If $y\in\mathbb{Y}$ is $(\mu_{a_0})_{T_{a_0}}$-computably random uniformly relativized to $a_0$, then there exists $x\in\mathbb{X}$ which is $\mu_{a_0}$-computably random uniformly relativized to $a_0$ such that $y=T_{a_0}(x)$. This is presented as the full generalization of the non-relativized result to uniform relativization; the surrounding discussion explains why uniform relativization is the relevant strengthening for computable randomness. The supplied text gives no resolution, so the conjecture remains open.