Lee's conjecture on the Frobenius norm

Let $M_n$ denote the space of $n\times n$ complex matrices, and let $|A|$ denote the absolute value of a matrix $A\in M_n$. Write $\|\cdot\|_F$ for the Frobenius norm.

Lee's conjecture. For all $A,B\in M_n$,

$$\|A+B\|_F\le \sqrt{\dfrac{1+\sqrt{2}}{2}}\left\|\lvert A\rvert+\lvert B\rvert\right\|_F.$$

Lee's conjecture asks for the optimal constant in the corresponding Schatten-norm inequality. The Frobenius-norm case was proved by Lin and Zhang, so this conjecture is solved.