Let $b: \mathbb R_+\times\mathbb R_+\times \mathcal P\to\mathbb R$ be Lipschitz, where $\mathcal P$ denotes the set of probability measures $\mu$ on $\mathbb R_+$ of finite first moment and is endowed with the Wasserstein metric of order $1$. Consider the equation

$$(1+\alpha)c(t,\mu, \alpha)=\int_{\mathbb R_+} \big(b(t,x,\mu)+c(t,\mu,\alpha)\big)^+\mu(dx) - (1-\alpha)\big(b(t,0,\mu) + c(t,\mu,\alpha)\big)^+,\quad\quad (\ast)$$

for all $t\in\mathbb R_+$, $\mu\in\mathcal P$ and $\alpha\in [0,1]$. My questions are as follows :

Does $(\ast)$ admit a unique solution $c: \mathbb R_+\times \mathcal P\times [0,1]\to\mathbb R$?

If so, could this solution be Lipschitz? If not, could this solution be Lipschitz w.r.t. $(x,\mu)$ and continuous w.r.t. $\alpha$?

PS : Some special cases have been studied, e.g. $b\ge 0$ or $b\equiv b(t,\mu)$, where we may derive the explicit expression for $c$. So my question is rather for the case where $b$ may change sign and depend on $x$.