I think this is the key. There are some language quirks, but Matlab has a very polished IDE experience with easy editing/debugging of scripts (note that I didn’t say functions) and a very responsive interface for plotting/etc. You never have to wait for anything… As far as serious programmers are concerned, the IDE is terrible (e.g. lack of standard code editor features like multi-cursors, etc.) but they are not the target audience.
So comparing Matlab to Julia may not be what they have in mind. What they are probably thinking of is Matlab vs. Julia+Jupyter or Julia+Atom. For that, it can’t compete with the polish of Matlab, but it is making good progress. If you are worried you only have one shot to convince them, you might want to wait until the plotting speed (and, depending on whether they write scripts or functions, some of the scoping things) make progress. Regardless, it may be helpful to set expectations on the degree of polish of the IDE experience relative to an expensive commercial alternative.
But saying “prototyping” usually means that you end up writing things in a different language, like Fortran. For that, I think Julia dominates matlab because you don’t need to prototype in different languages. If that is the goal, and matlab is not high enough performance to solve their problems, then any warts in the Julia IDE are worth the trouble in not having to then move to another language for the “real” problem.