What are 6 colors which are also well-distinguishable in grayscale?

Well. In reality your colors could work well. The problem you have is who is converting the colors.

Your grayscale image looked totally strange to me because the yellow should be printed very light.

Here is your image converted to grayscale with a simple editing program. (The black background is just because it is a transparent png)

There is a chance that whatever “software step” it is converting your image is making a strange decision.

It could be a printer driver, or something.

There is a chance that that driver is only taking in acount saturation. All colors look saturated. Yellow, magenta, cyan, green, except the pink that could be an unsaturated red.

You could try to use an unsaturated yellow (pastel yellow), the unsaturated red( pink), a saturated cyan, etc.

But probably what you really need is to make diferent versions or the file if you are using a non profesional program for making your publication, and you need to stick with it.

If you are using any profesional program the color mode conversion will be more predictable. You only need to export to a PDF with a grayscale color output.

You could use: Scribus https://www.scribus.net/, Page Plus Starter edition http://www.serif.com/desktop-publishing-software/ which are free.

But even Word makes a better color conversion, this is a pdf generated from PDF creator, which is a driver to make a pdf printing from any aplication. It is pixelated because I just made a quick conversion, but the grays are better.

In any case, you could print this kind of images: https://www.google.com/search?q=rgb+color+swatches and choose some that fits your overall design in color and in grayscale. But again, I would use another program to prepare my publication.