Symmetries of effective field theory of hydrodynamics: a confusing calculation

This is a very specific question about a paper by P. Glorioso and H. Liu that can be found here https://arxiv.org/pdf/1805.09331.pdf. In particular I want to understand how the authors get from the Lagrangian (6.71) to the constrained symmetry (6.72).

BACKGROUND:

The set up is as follows: We have four scalar fields $$\sigma^A(x)$$, ($$A=0,1,2,3$$) living in a 4D spacetime. The fields $$\sigma^i$$ for $$i=1,2,3$$ correspond to fluid elements and the field $$\sigma^0$$ can be thought of as co-moving, internal clocks for each fluid element. The action has the following symmetries

1. Fluid element re-labeling: $$\sigma^i\to\sigma'^i(\sigma^j)$$ for $$i,j=1,2,3$$ and $$\sigma'^i$$ is any diffeomorphism of $$\sigma^i$$.

2. Re-parameterizations of the 'internal clocks' $$\sigma^0\to \sigma^0+\alpha(\sigma^i)$$ for generic infinitely differentiable function $$\alpha$$.

At lowest order in derivatives, the action expressed in physical space-time takes the form $$S=\int d^4 x \sqrt{-g}~P(b),$$ where $$b\equiv-g_{\mu\nu}(K^{-1})^\mu_0 (K^{-1})^\nu_0$$ and $$K^A_\mu\equiv \partial_\mu\sigma^A$$ and $$P$$ is an arbitrary function (that can be specified by experiments). [Note: in the paper they have a different form of the action given in equation (6.71) defined on the fluid corrdinates $$\sigma$$. The action I have written down above is the same action but expressed as an integral over physical space-time coordinates $$x$$.]

The authors argue starting from the above action, it is possible to integrate out the $$\sigma^0$$ fields to obtain an action for just the $$\sigma^i$$ fields. They then claim that "When solving $$\sigma^0$$ in terms of other variables, one finds an arbitrary integration function, which can be fixed to be unity and in turn breaks the [fluid element re-labeling] symmetry down to volume-preserving spatial diffeomorphisms" $$\sigma^i\to f^i(\sigma_j)~~~~~\det\bigg(\frac{\partial f^i}{\partial \sigma^j}\bigg)=1.$$

QUESTION: I am having trouble carrying out the computation of integrating out the $$\sigma^0$$ fields to obtain the volume-preserving spatial diffeomorphism symmetry. Can someone please show me how to do this computation explicitly and in detail?