This work explores the asymmetry of quantum steering in a setup using high-dimensional entanglement. We construct entangled states with the following properties: (i) one party (Alice) can never steer the state of the other party (Bob), considering the most general measurements, and (ii) Bob can strongly steer the state of Alice, demonstrating genuine high-dimensional steering. In other words, Bob can convince Alice that they share an entangled state of arbitrarily high Schmidt number, while Alice can never convince Bob that the state is even simply entangled. In this sense, one-way steering can become unlimited. A key result for our construction is a condition for the joint measurability of all high-dimensional measurements subjected to the combined effect of noise and loss, which is of independent interest.
