$\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$It follows from Proposition 7 and this recent answer that, for any positive-definite $n\times n$ symmetric real matrices $A$ and $B$,

$$\tr\big[A+B-2(A^{1/2}BA^{1/2})^{1/2}\big]\ge
\Big(\frac{\|A-B\|}{\sqrt{\|A\|}+\sqrt{\|B\|}}\Big)^2, \tag{1}\label{1}$$
where $\tr$ denotes the trace and $\|M\|$ is the spectral norm of a matrix $M$.

That proof of \eqref{1} involves certain probabilistic arguments.

Since \eqref{1} is stated in purely matrix terms, the following question naturally arises:

Is there a proof of \eqref{1} involving (no probabilistic tools, but) only matrix analysis tools?

A related question:

Will inequality \eqref{1} hold (perhaps up to a universal positive real constant factor) if each instance of the spectral norm in \eqref{1} is replaced by that of the Frobenius one?

Any correct and complete answer to either one of these two questions will be considered a correct and complete answer to this entire post.