The NHS website makes no mention of milk or sugar under 'triggers':
https://www.nhs.uk/conditions/psoriasis/causes/
The Psoriasis and Psoriatic Arthritis Alliance's website says this:
"A healthy diet is important for wellbeing and can reduce your risk of many long-term illnesses. However, there is no clear link between what you eat and the severity of psoriasis symptoms"
https://www.papaa.org/learn-about-psoriasis-and-psoriatic-arthritis/just-diagnosed/what-is-psoriasis/
(See also
https://www.papaa.org/learn-about-psoriasis-and-psoriatic-arthritis/further-information/psoriatic-lifestyle-and-nutrition )
Perhaps more tellingly, the Psoriasis Society's website says this:
"If you look online, you’ll find lots of diets that ‘cure’ psoriasis, and you might see features in magazines or hear people saying that psoriasis can be controlled through eating - or not eating - certain foods. The truth is that scientific research has not yet found a definite link, or found a diet that works for everybody. Some people do say that they’ve noticed an improvement when they cut something out of their daily diet, and if you think you might be eating something that makes your psoriasis worse, then keeping a food diary is an easy way to spot any patterns"
https://www.psoriasis-association.org.uk/psoriasis-and-treatments/faq