The grounds that they are putting forward (as far as I can make them out) are:
1 The court was wrong to say that the statement "If trans women are permitted to use a single-sex female lavatory all biological males must be permitted to use that lavatory" is legally correct as the use of the word "must" implies that it will always be the case.
However, the court said in the judgment at [59]-[61] that it will depend on the circumstances.
GLP are saying that you can't have it both ways. A statement that uses the word "must", must indeed always apply, not just depending on the circumstances.
[to my mind, they may have an argument here]
They also go on to make the same claim that the WI did, that this is about "living as women" rather than being an actual man or woman and so a non trans-identifying man can lawfully be excluded.
[I'll be interested to see where that argument goes, as this was what the WI argued against my DH]
They also argue that the court failed to properly consider their argument that providing trans-inclusive toilets was positive action under Section 158 Equality Act.
[I don't think that will likely go anywhere]
Further, the update didn't fully address the risk of gender reassignment discrimination.
They go into a lot of detailed argument about why excluding trans people form toilets is both direct and indirect gender discrimination.
GLP argue, among other things, that providing mixed-sex facilities would be outing to trans people and thus indirect discrimination. They also argue that the correct comparator for a trans-identifying man should still be a woman.
.
2 "Failure properly to construe the Interim Update in accordance with its natural meaning"
They seem to argue that the Interim Update assumed that they were talking to an employer or service provider that had at least some nodding acquaintance with discriminaiton law or they have legal advisors.
They argue that this should not be assumed and that there is a "natural and ordinary meaning" to the words.
They say, that an ordinary employer or service provider would read the Update and consequently may require a trans person to use the facilities designated for their “biological sex” if they do not have any mixed-sex facilities available.
They then argue that para [69] of the judgment is wrong and that:
"It follows that, where for practical reasons it is only possible to provide toilets designated for men and women, trans people may be required to use the toilets designated for their sex at birth. The Judgment does not engage anywhere with the reality that a proportion of duty bearers will not have the resources or facilities to offer additional spaces."
Which would be a bad thing.
They also argue that what was said about toilets in the Workplace Regs in the judgment at [34] is also not correct.
.
3 "Erroneous construction of the 1992 Workplace Regulations"
Again, they are repeating the argument from the original case that this can be done on a "trans-inclusive basis" and the court was wrong to say otherwise.
They have a very long argument as to why they are correct and the court isn't.
.
4 "Erroneous approach to Article 8 ECHR"
They then go on to a very long argument as to why there would be a breach of Article 8 rights.
.
Personally, I don't think most of these will stand up to scrutiny (but there again, what do I know about anything?)