Why this problem is unbounded, even though it's dual is feasible?

Question

In this paper, An Efficient Inexact ABCD Method for Least Squares Semidefinite Programming,page 2, there is a problem called, (P): \begin{align} P: &\min_{X,s} && ‎ \frac{1}{2} ‎\Vert X-G\Vert^2‎ +\frac{1}{2} \Vert s-g\Vert^2 ‎\cr‎ &\text{s.t.}\quad ‎‎‎‎‎&& ‎A_{E}(X)= ‎b_{E}\cr‎ &&& ‎A_{I}(X)= s, X\in S^n_+, X\in \mathcal{P},s\in \mathcal{K} \end{align}‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎ And then defined problem D, it's dual as: \begin{align} D: &\min_{Z,v,S,y_E,y_I,} && -\langle b_E,y_E\rangle+\delta_{S_+^n}(S)+‎\delta_{\mathcal{P}}^*(-Z)+\delta_{\mathcal{K}}^*(-v)+ \cr&&&\frac{1}{2} ‎\Vert A_E^*y_E+A_I^*y_I+S+Z+G\Vert^2+\frac{1}{2} ‎\Vert g+v-y_I\Vert^2-\frac{1}{2} \Vert G\Vert^2-\frac{1}{2} \Vert g\Vert^2 \cr&&&\delta_{\mathcal{C}}^*(.) = \sup_{W\in \mathcal{C}} \langle . ,W\rangle \end{align}‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎‎ Is this dual correct? If it is why, mosek,sdpt3, says it's primal infeasible(unbounded)? what is the constraint to impose $\delta_{\mathcal{P}}^*(-Z)$ and $\delta_{\mathcal{K}}^*(-v)$ in non-projective manner as paper says it's done?