Postnikov invariants of the Brauer 3-group Given a commutative ring $k$ there is a bicategory with 


*

*algebras over $k$ as objects,

*bimodules as morphisms,

*bimodule homomorphisms as 2-morphisms.  


This is a monoidal bicategory, since we can take the tensor product of algebras, and everything else gets along nicely with that.
Given any monoidal bicategory we can take its core: that is, the sub-monoidal bicategory where we only keep invertible objects (invertible up to equivalence), invertible morphisms (invertible up to 2-isomorphism), and invertible 2-morphisms.  
The core is a monoidal bicategory where everything is invertible in a suitably weakened sense so it's called a 3-group.    
The particular 3-group we get from a commutative ring $k$ could be called its Brauer 3-group and denoted $\mathbf{Br}(k)$.  It's discussed on the $n$Lab: there it's called the Picard 3-group of $k$ but denoted as $\mathbf{Br}(k)$.
Like any 3-group, $\mathbf{Br}(k)$ has homotopy groups which I will call $\pi_1, \pi_2, \pi_3$ (though there are choices of where we start numbering).  These are well-known things:


*

*$\pi_1$ is the Brauer group of $k$.

*$\pi_2$ is the Picard group of $k$.

*$\pi_3$ is the group of units of $k$.
My question is whether people have studied, or computed, the Postnikov invariants involving these things.  The simplest is the map
$$ a : \pi_1^3 \to \pi_2$$
coming from the associator in the monoidal category of $k$-algebras (with isomorphism classes of bimodules as morphisms).   Since the associator obeys the pentagon identity this is a 3-cocycle on $\pi_1$ with values in its module $\pi_2$, so it gives an element of $ H^3(\pi_1, \pi_2)$.
Is this element trivial?  If not, what is it?   
But in fact $\mathbf{Br}(k)$ is not just a 3-group but also a symmetric monoidal bicategory.  So, it's what I call a symmetric 3-group, though some others call it a Picard 2-category.  These have a number of other Postnikov invariants:


*

*Nick Gurski, Niles Johnson, Angélica M. Osorno and Marc Stephan, Stable Postnikov data of Picard 2-categories.


Has anyone figured out any of these for $\mathbf{Br}(k)$?
 A: Let me see if I understand what Jacob says in the comments. I think his argument can be summarized as: the Brauer 3-group is étale-locally an Eilenberg-MacLane spectrum, hence étale-locally an $\mathbb{Z}$-module spectrum, hence an $\mathbb{Z}$-module spectrum, hence the Postnikov tower splits. Do I have that right?
If so, I want to point out that the situation changes with one tweak, which is to allow superalgebras, super bimodules, etc. When $k = \mathbb{R}$ the super Brauer 3-group has a nontrivial homotopy operation $\pi_1 \to \pi_3$ given by taking the super dimension of the zeroth Hochschild homology of a superalgebra in the super Brauer group, and I computed an example of it taking a nontrivial value here. This implies that the Postnikov tower can't split but I don't know what the $k$-invariants are. I suppose following Jacob we could try to work things out as homotopy fixed points of the $\text{Gal}(\mathbb{C}/\mathbb{R})$-action on the super Brauer 3-group over $\mathbb{C}$ but this is beyond me.
