Let $\kappa>0$ and $d$ be a positive integer. For $k\in \mathbb{Z}^+$ large enough, can one find a $d$-dimensional geodesically complete and simply connected Riemannian submanifold $(M_{\kappa},g)$ of the $k$-fold product of the hyperbolic plane $\prod_{i=1}^k\, \mathbb{H}^2$ with sectional curvature bounded in $[-\kappa,0]$?

Ideally, can we take it to have constant sectional curvature $-\kappa$?

Let $\kappa>0$ and $d,k$ be a positive integers with $k\ge d$. For $k\in \mathbb{Z}^+$ large enough, can one find a geodesically complete and simply connected $d$-dimensional Riemannian submanifold $(M_{\kappa},g)$ of the $k$-fold product of the hyperbolic plane $\prod_{i=1}^k\, \mathbb{H}^2$ with sectional curvature bounded in $[-\kappa,0)$?

Ideally, can we take it to have constant sectional curvature $-\kappa$?