Question about a paper of Longuet-Higgins on the steepest progressive wave in deep water

I have great difficulties to understand the article "On the Form of the Highest Progressive and Standing Waves in Deep Water", which is available freely on Jstor. This article deals with the largest possible steepness, $(H/lambda)_{max}$, of a stable progressive wave in deep water. Stokes had shown in 1880 that waves at breaking point (i.e, of largest steepness) have crests with cuspidal shape, with opening angle $120$ degrees. This part of the theory of breaking waves in deep water in totally clear to me. Higgins in the paper previously cited uses a certain mathematical trick to derive the profile of a full cycle of Stokes sharpened waves, and as a by product obtains the value $frac{ln(2/sqrt{3})}{pi/3}approx 0.1374$ for the maximal wave steepness, a value which is very close to the results of previous studies, which were done from more "physical" viewpoint.

His trick is essentially to look at the fluid domain bounded by six consecutive full waves cycles as a domain in the complex plane , and to use the conformal mapping $z’ = e^{-iz}$ to map this fluid domain to a domain in the $z’$ plane. In this way, $y = -infty$ (infinite depth) is mapped to the origin of the $z’$ plane, lines parallel to the $x$ axis to circles around the origin, and lines parallel to $y$ axis to emanating rays from the origin. Since this mapping is conformal, it conserves the $120$ degrees angles, and thus, Higgins argues, the six crests of the waves are mapped onto the vertexes of a regular hexagon. He than states that the profile of one wave cycle (from crest to crest) is mapped to a straight line connecting two adjacent vertexes of the hexagon, and along this reasoning derives the shape of one wave cycle: $y = ln (sec x)$ (where $x = 0$ at the trough of the wave).

The main points I don’t understand are:

• Higgins’s argument implicitly assumes that the profile of the wave of maximal steepness must be convex. Why is it so?
• Even if the first point is true, it still doesn’t imply that one wave cycle is mapped to a straight line, as one can construct a cyclic hexagon and connect the vertexes with arcs that intersect the radiuses at angles $60$ degrees. In Higgins’s article, it seems somehow obvious that the fluid domain is mapped to a regular hexagon, and i don’t understand the methodological principle which implies that for steepest waves with crest angle $theta$ one should assume that the conformal mapping $z’ = e^{-iz}$ sends their profile to the sides of a regular polygon of internal angles $theta$.

Update: It’s important to remark that Higgins’s trick is intended to be an approximation for the profile of steepest wave, but i don’t understand at all how the "hexagon approximation" can be concieved to be a first-order approximation, and obviously i don’t see how can one see intuitively that such a trick should work.