Lockability and non-lockability are explained in this paper. A real valued function of a quantum state is called non-lockable if its value does not change by too much after discarding a subsystem. The max-entropy of a quantum state is defined as
$$H_{\max }(A)_{\rho}= \log \operatorname{tr}(\rho_{A}^{1 / 2})$$
For a bipartite quantum state $\rho_{AB}$, I would like to know if the max-entropy is non-lockable i.e. is there any relationship of the form
$$\text{tr}(\rho_{AB}^{1/2}) \leq \text{tr}(\rho_{A}^{1/2})\cdot|B|$$
which, after taking logs on both sides, would yield
$$H_{\max}(AB)\leq H_{\max}(A) + \log|B|$$
A couple of easy numerical examples suggests this might be true but I have not been able to prove it.