the ratio of conductivity/conductivities of two phases
Solution 1:
TLDR: I think you are right, and what Edwin said.
In formal writing, ratios should be explicitly described as something to something else. So what you should say is something like the ratio of the conductivity of one phase to that of the other. For a bigger sample (12,000), I swapped density for conductivity, and Google Scholar spat out the following for "ratio of the density of" -
Sgro (1975) has found that there is a characteristic ratio of the density of the material inside the cloud to that outside the cloud such that for small density ratios the cloud will be left heated and expanded, but for large density ratios the cloud will be left cooled and compressed
The supernova trigger for formation of the solar system AGW Cameron, JW Truran - Icarus, 1977 - Elsevier -https://www.sciencedirect.com/science/article/pii/0019103577901014
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The weakness of the processes H + p - H~ + hv and H + H - Hz + hv indicates the negligible ratio of the density of Hz to that of H in the intergalactic medium.
[PDF] Recombination of hydrogen in the hot model of the universe -http://jetp.ac.ru/cgi-bin/dn/e_028_01_0146.pdf
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where o is the ratio of the density of the lead ahead of the shock wave to that behind.
Further results in the theory of long rod penetration∗ - https://www.sciencedirect.com/science/article/pii/0022509669900283
I found no examples in the first few pages of results that were constructed the way your second example is.
Changing the search to densities, as in "ratio of the densities of", yields a high proportion of results structured like your first example (out of 3300 hits)-
For not very high primary energies, one would have to multiply this ratio by a ratio of the densities of the final electron states to obtain the actual cross-section ratio
On the long-range correlation model of the photonuclear effect - https://www.sciencedirect.com/science/article/pii/0029558258900051
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it is clear that supercritical releases of different densities should give decay constants k, whose ratio is equal to the reciprocal square root of the ratio of the densities of the fluids.
The structure and concentration decay of high pressure jets of natural gas - https://www.tandfonline.com/doi/abs/10.1080/00102208408923739
I can't think of any reason why conductivity and density ought behave differently here. They are both bulk properties of materials. I do think the first example is a bit casual unless you expect the reader to already be familiar with the terms. Note that neither example above is using that construction to introduce new ideas, only to expand on earlier remarks using already familiar terms.