A short proof of the modified Kretschmann-Schlingemann-Werner conjecture

Let ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2:{\mathbb{M}}_d(\mathbb{C})\to {\mathbb{M}}_n(\mathbb{C})$ be two quantum channels with respective Stinespring isometries $V_1,V_2:{\mathbb{C}}^d\to {\mathbb{C}}^n\otimes {\mathbb{C}}^m$ on any common dilation space ${\mathbb{C}}^m$. We prove that there exists a unitary $U$ on ${\mathbb{C}}^m$ such that $\mathrm{\parallel }{V}_1-(\mathbf{1}\otimes U)V_2{\mathrm{\parallel }}_{\mathrm{\infty }}\le \sqrt{2\mathrm{\parallel }{\mathrm{\Phi }}_1-{\mathrm{\Phi }}_2{\mathrm{\parallel }}_{\diamond }},$ thus resolving vom Ende’s modification of the Kretschmann-Schlingemann-Werner conjecture in the affirmative.