SDP problem with Schur complement-like constraint

Question

I'm interested in what we can conclude about the following question. Let $\mathbf{Y} \in \mathbb{R}^{n \times n}$ be in the convex hull of projection matrices of rank at most $k$ (equivalently, $\mathbf{0} \preceq \mathbf{Y} \preceq \mathbb{I}_n$ and $\text{trace}(\mathbf{Y}) \leq k$), and let $\mathbf{Z_2} \in \mathbb{R}^{n \times k}$ be known. Is there an "easy" way to solve: \begin{align*} \min_{\mathbf{Z_1}, \mathbf{Z_3}} \quad & \langle \mathbf{Z_1}, \mathbf{Y} \rangle + \langle \mathbf{Z_3}, \mathbb{I}_k \rangle \\ \text{s.t.} \quad & \begin{pmatrix} \mathbf{Z_1} & \mathbf{Z_2} \\ \mathbf{Z_2}^\top & \mathbf{Z_3} \end{pmatrix} \succeq \mathbf{0} \\ & \mathbf{Z_1} \in \mathbb{R}^{n \times n}, \ \mathbf{Z_3} \in \mathbb{R}^{k \times k} \end{align*}