AFAIK, non-signed crowns are the norm in 1958. The JL logo was introduced in literature around that time, and crowns were signed on most models released only afterwards.
For example E853 was born with non-signed crowns in 1957, and so were the 3159, 3160 and 3161 in 1958.
You might call it a rumor, but to me there is empirical evidence that there was a transition to models with signed crowns introduced at about the release of the case serial number 760,000 (+/- 10,000), and therefore the E855, E856 / E858 and later E's have signed crowns.
With regards to the E858 pictured above, I highly doubt it is franken. It is mine, part of a collection of 5 858, including 2 parkings, of which one Gübelin, and I think I can tell a franken apart, especially given the very specific size of the alarm dial on that reference.
In my book, there are 3 correct P references: 3161 / E856 / E858, the latter with both Jaeger LeCoultre and Gübelin versions.
Now, back to ref 3161, Nicolas has shown some steel and yellow gold. How about some Pink ?