Their letter starts off with a whole load of waffle about the history of trans rights cases.
Then, by the time you get to page 16, it gets on to the main issue.
The GLP point out that the interim guidance says
"In workplaces, it is compulsory to provide sufficient single-sex toilets, as well as sufficient single-sex changing and washing facilities where these facilities are needed."
But the Workplace Regulations 1992 say
"separate rooms containing conveniences are provided for men and women except where and so far as each convenience is in a separate room the door of which is capable of being secured from inside."
Or, in other words, separate self contained toilets used by one person at a time are a suitable alternative to single sex toilets.
They seem to think that this is a huge "gotcha".
But they do go on to say that since this is done under the Workplace Regs and not the Equality Act then the FWS case has no effect when it comes to toilets in the workplace.
As a result, they say, it is necessary to look to the GRA 2004 and that people with a GRC should still be able to use toilets of the opposite sex in a workplace.
They then go on to explain some way in which service providers could also allow trans people into public toilets of the opposite sex if they wished to.
I have no idea how the Equality Act might interact with the Workplace Regs (or if the two even do interact) but this is the argument that GLP are putting forward.
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The second ground of their claim is that since they believe that the interim advice is wrong that issuing it has been a breach of the Human Rights Act and so should be withdrawn.
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The third ground is that if the interim guidance is correct then the law is incompatible with the Human Rights Act.
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Oh, and by the way, they also want:
"The Commission is requested to withdraw its Interim Update immediately and to state publicly that the Interim Update is of no effect and its advice should not be followed."
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How strong a case or otherwise this is, I have no idea. But this is their latest move.