This took me multiple hours to figure out because the solution polylang provides by itself doesn't work here or with Lay Theme:
If you have the problem that your homepage doesn't switch the language but all other pages work. Like, German is your main language and English your second language, the switch always shows the German translation – unimportant what you do:
Do NOT set the homepage under the ‘Reading settings’ panel.
Leave it as is and set your Lay Theme homepage via the usual way in the Customize panel.
Then it works!
PS: It also works with any page being selected in the ‘Reading settings’ – just don't select the actual homepage. The Lay Theme setting overrules that setting anyways, but if you set both ways with the same page, it breaks polylang for the homepage.