oc.optimization and control – Proximal operator of composition with norm squared on a Hilbert space

Let $(X,Sigma,mu)$ be a finite measure space and let $L^2(mu,mathbb{R}^d)$ denote the Bochner space of strongly measurable functions taking values in $mathbb{R}^d$. Let
T &:L^2(mu,mathbb{R}^d)rightarrow L^2(mu,mathbb{R})\
& fmapsto |f|_2,

where $|cdot|_2$ is the Euclidean norm on $mathbb{R}^d$. Fix a $phiinGamma_0(L^2(mu,mathbb{R}))$.

Is there a known expression for
operatorname{Prox}_{Tcirc phi}

in terms of $operatorname{Prox}_{phi}$, $phi$, and $T$?