Dimension-free maximal symmetric modulus conjecture

For $Z\in\mathbb{M}_d$, define the maximal symmetric modulus by

$$|Z|_{\vee}=|Z|\vee|Z^*|.$$

For $A,B\in\mathbb{M}_d$, consider whether there are unitaries $U,V\in\mathbb{M}_d$ and a constant $c_{\vee}(d)$ such that

$$|A+B|_{\vee}\le c_{\vee}(d)\left(U|A|_{\vee}U^*+V|B|_{\vee}V^*\right).$$

Maximal symmetric modulus conjecture. For every dimension $d$, one may choose $c_{\vee}(d)=1$. The assertion would settle the corresponding finite-dimensional question in its strongest proposed form; it is not resolved in the supplied text.