The proper ratio inside the coaxial connector for Teflon is correct OD/ID = 3.3.
What you showed was the 2D interface wall which was misleading.
The ratios for 50 Ohms with a pin on the top layer with a ground plane and coplanar grounds cannot be given externally, unless Dk is known, except for the air above the surface Dk = 1 to 1.01(?)
(Edit)
For FR-4 50 ohms has W/H = 1.8 but this task needs a very low loss tangent substrate like PTFE in the PCB where W/H = 3.0 which requires a thinner substrate.
It is interesting to note on the PCB that this coplanar waveguide gap to trace on the top surface has low sensitivity to trace impedance. Yet it has high effectiveness for parasitic shielding for mm wavelength noise.
It's hard to show a 3D problem in 2D. Even this illustration I modified needs more info.
Some may wonder how the pin contact works with such a small contact area gold to gold on a good ultra-smooth PCB between 1 and 18 GHz.
This is why contact force is so important with a Au-berrylium-copper flexible pin and connector frame & bolts fit for a tank for a precision elastic force to give a gas-tight fit. The PCB must not deform from the pin, as I may have implied in order to be repeatable for contact reliability long-term.
It appears that the trace shows a short taper widening the PCB trace width. However simulations must determine this so that sensitivity to the wave impedance does not change significant. With an abrupt change from 3D coax to 3D rectangular waveguide using the same PTFE, the ratios will change abruptly so it is not clear how much if any taper is needed. The trace is very thin compared to the pin, yet most of the pin is air gapped so the L/C ratio is not what you might expect in the radial to orthogonal geometry change. An EM simulation will be necessary with Maxwell or Ansys or equiv.