I can see how it follows from $\sqrt{x} \geq \sqrt{a} +\frac{x-a}{2\sqrt{a}} - \frac{(x-a)^2}{2a^{3/2}}$, which Mathematica tells me is true for all $x \geq 0$ and $a>0$. I suppose there is some simple translation from the inequality you wrote to this more general one? If you consider this too simple or obvious, please don't bother...

I don't see how the general inequality $\sqrt{E(X)}\bigg(1-\frac{Var(X)}{2 E(X)^2}\bigg) \le E(\sqrt{X})$ for an arbitrary non-negative RV $X$ follows from $1 + \frac{x-1}{2} - \frac{(x-1)^2}{2} \le \sqrt{x} $. Could you please clarify this point? Is it obvious?

There has now been even more progress on Conjecture 2 that is useful enough for applications in quantum information theory. The idea was to make use of the Hadamard three-line theorem (in particular, Riesz-Thorin interpolation). This is detailed in the following paper: arxiv.org/abs/1505.04661 . See also arxiv.org/abs/1506.00981 for follow-up work.

This is just to say that there has been what I would consider siginificant progress on Conjecture 2 since it was posted. Fawzi and Renner have proved a variation of the case when $\beta = 1$ and $\alpha = 1/2$, in the paper arxiv.org/abs/1410.0664 . My coauthors and I have also generalized the notion of Renyi conditional mutual information Renyi relative entropy differences in the paper arxiv.org/abs/1410.1443 and we generalized the methods of Fawzi and Renner in the paper arxiv.org/abs/1412.4067 . Of course, the conjectures stated here still remain open.