Operator maps of Jensen-type

Let $\mathbb{B}_J(\mathcal H)$ denote the set of self-adjoint operators acting on a Hilbert space $\mathcal{H}$ with spectra contained in an open interval $J$. A map $\Phi\colon\mathbb{B}_J(\mathcal H)\to {\mathbb B}(\mathcal H)_\text{sa}$ is said to be of Jensen-type if $$\Phi(C^*AC+D^*BD)\le C^*\Phi(A)C+D^*\Phi(B)D$$ for all $A, B \in B_J(\mathcal H)$ and bounded linear operators $C,D$ acting on $\mathcal H$ with $C^*C+D^*D=I$, where $I$ denotes the identity operator. We show that a Jensen-type map on a infinite dimensional Hilbert space is of the form $\Phi(A)=f(A)$ for some operator convex function $f$ defined in $J$.