How to create a nozzle removing material with variable section sweep?

Hi!
I need some help on below:
I am working on a plastic casing that has two main components as shown in the very first picture. There is already an inner cavity (turbine shaped cut to accept water) that ends to a rectangular opening. Client asked me to add material and extend this nozzle so that in the next 15cm to turn into a circle or ellipsis of same area.
In other words, we need to cut a almost tubular nozzle, which starts as rectangular and ends smoothly as circle or ellipsis.
Pls see in red above, the rectangular surface where removing material should start.

Since the ELLIPTICAL OUTLET OF NOZZLE should be created by the assembly of the two parts, we need half of the cut at each part.
That is why I have sketched a curve on the surface that we be removed to have it a refererce.

I tried Variable Section Sweep , with two trajectories that lead from the rencangular to the hald-ellipsis shape and I sketched a half ellipsis section.
And I get this:

which of course is not variable section cut, but only one section.

This cannot give me a smooth trancation for rectangular to elliptical opening.
Any advise or amendment to my thought?
I am not very familiar with variable section sweeps but I think there might be an easy way that I just dont know.
Please help me , the soonest the better. I m in despair! 
Thanks,
Amanda
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