  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.
