Brauer groups of Fano 3-folds

After returning from holidays earlier this summer, I organized all the stuff on my desk a bit, and I ran into various half-finished computations and notes. Whilst some might eventually become (short) papers, most of them will (at least initially) just be fun blog posts. The first topic I want to tackle is the Brauer group of Fano 3-folds. In the wonderful paper Fano varieties with torsion in the third cohomology group by John Christian Ottem and Jørgen Vold Rennemo, they construct even-dimensional Fano varieties (from dimension 4 onwards) for which $\mathrm{H}^3(X,\mathbb{Z})\cong\mathbb{Z}/2\mathbb{Z}$. For a Fano variety, the torsion subgroup of $\mathrm{H}^3(X,\mathbb{Z})$ is its cohomological Brauer group. For a del Pezzo surface there is nothing to be said about (torsion in) $\mathrm{H}^3(X,\mathbb{Z})$: we have that $\mathrm{H}^3(X,\mathbb{Z})=0$. But a priori there could be something interesting happening in dimension 3, as $\mathrm{H}^3(X,\mathbb{Z})$ can certainly be…