Reference prior for a three-parameter model and likelihood factorization

Let a (regular) statistical model with three parameters $\phi_1$, $\lambda_2$, $\mu$, and three observations $x_1$, $x_2$, $y$. Assume the likelihood has form $$L(\mu,\phi_1,\lambda_2 \mid y, x_1, x_2) = L(\mu,\phi_1 \mid y, x_1) \times L(\lambda_2 \mid x_2).$$ Consider any ordering of the parameters with $\phi_1$ as the main parameter of interest. Can we deduce from the properties of (Berger & Bernardo's) reference priors that :

• the marginal reference posterior distribution of $\phi_1$ only depends on $y$ and $x_1$ ?

• this reference posterior distribution of $\phi_1$ is the same as the one in the model obtained after removing $x_2$ from the observations ?

Assuming a positive answer to the first question, the second one looks like something close to the so-called consistency under marginalization. But not exactly. Consistency under marginalization, as I understand it from its statement given here (p6) would rather say: if $\pi(\phi_1 \mid x_1, x_2, y)=\pi(\phi_1 \mid x_1, y)$, and if the law of $(x_1,y)$ does not depend on $\mu$ and $\lambda_2$, then the desired property of question 2 holds true. But here the law of $(x_1,y)$ depend on $\mu$.