Can ring inversion in Cis-Decalin allow the E2 reaction?

This is a tutorial question I need to do, I initially thought the question was testing whether students properly understand (1) chair conformations of decalin, and (2) that the leaving group and the breaking bond should be 'anti-periplanar'.

However, the third molecule also turned out to have undergone the elimination reaction when the first didn't, so I am very confounded. I have drawn all possible conformations of cis-Decalins, but no single conformation seems to allow $$\ce{C-OTs}$$ bond and $$\ce{C-OH}$$ bond to be anti-periplanar. Am I missing out something?

The issue in this Grob fragmentation is whether or not the $$\ce{C-C}$$ and $$\ce{C-OTs}$$ bonds are anti-periplanar. The $$\ce{C-OH}$$ bond is free to rotate and the unpaired electrons of the negative oxygen can align with the other two bonds. In the first example a), the red bonds are gauche to one another. Standard E2 elimination is likely the source of the mixture. Examples b), c) and d) all have red bonds anti-periplanar. The blue bonds predict the geometry of the double bonds in the product. Examples b) and c) give E-alkenes while d) affords a Z-alkene.