How do I create this color-picker-HSB gradient?

What seemed trivial at first glance has turned out as quite a challange!
I would like to create a special kind of black/white/alpha gradient to mimic the HSB color picker in Photoshop:
Saturation
0 100
+--------------+ 100
|white.....pure hue|
|.....................| B
|.....................| r
|.....................| i
|.....................| g
|.....................| h
|.....................| t
|.....................| n
|.....................| e
|.....................| s
|.....................| s
|.....................|
|.....black......| 0
+--------------+
I've tried numerous ways of blending screen/multiply, setting gradient smoothing to 0-100%, using layer masks, etc, etc - but when I compare my composite with a screenshot of the Photoshop color picker - I'm still not even close! :-(
Is the gradients non-linear? Is gamma involved?
I downloaded a low-res png-24 of this unique gradient from the internet, but would like to make a hi-res version (and understand how it works!).
Additionally, my layer mask isn't black white but black and pink! I can't wrap my head around that color management thing...
Best,
Robert

Photoshop has not given us a tool to produce gradients in HSB (only RGB). In order to get the color picker slices it does produce, Photoshop itself is obviously able to create HSB gradients.

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