The weak Dirichlet problem for Baire functions

Let $X$ be a simplex and $K$ a compact subset of the set of all extreme points of $X$. We show that any bounded function $f$ of Baire class $\alpha$ on $K$ can be extended to a function $h$ of affine class $\alpha$ on $X$. Moreover, $h$ can be chosen in such a way that $h(X)\subset \overline{\operatorname{co}} \,f(K)$.