I don't think this part of the quest is written yet. Actually, it seems that Vandenberg's wife doesn't allow lewd interactions now apart from just groping. Got her at 100 slutiness and 100 love, can't even trigger the scene that I encountered before where she asks you to stay for dinner. So for the time being, I'd say this quest is on hold.
I got it to trigger while still tutoring Emily at night. Once you hire her or she passes her exam, this is not doable. But I feel like there's random bugs. My Sakari quest for example is not triggering, maybe because I oneshotted the payment about curing her illness before Kaya joined the internship. There's some random hints that do not do anything too (in the current playthrough).
Inspection visit is determined (in this case) before accepting the chemist daughter quest, so reloading to save after accepting the quest don't work which is just bad design. Since inspection is set to stone, stopping production, hanging around in x or anything else wont fix this. Even cheating and setting Attention to 0 after getting the quest wont work. Only fix seems to be to go trough the inspection and then use cheats to set day time back to afternoon.
Oh wow so that's whap happened with me. I don't think I have a valid save and I do not know the command details to check quest status or set triggers.
I was trying to get a run with minimal hired employees (mostly only story employees except for one just to try to trigger a side quest in particular) and didn't know why side quests were never starting lol.
For those wondering for how I financially survive a minimal employee run (as a somehow tedious challenge), the sex shop is available very early, and you start with some money:
- If you start investing your start money on the sex shop, it is possible to offset the early salaries.
- Use performance review (and punishments later but maybe not needed anymore when available) to decrease the salaries of your staff. High charisma stat allow to decrease salary without affecting obedience.
- Use compliment work to keep them happy, try to get them a high obedience, or pleasure to convince them to stay
- Groping alone can raise sluttiness to 100, and is safe to use in public (police aside). That way you can kickstart a lot of sluttiness/happiness/obedience for as many employees as you want (need obedience to get them to follow though). You can even take a girl you can grope to follow you to a place with someone you cannot grope and/or does not like you, grope in front of her to increase the sluttiness beyond flirting if you want to squeeze out your energy. Public sex is a weird (abusable) mechanic, like you can creampie your sister in front of your mom, they will be ok with it, but storywise would still be worried about dating.

- Strategy is then to max out the foreplay skill to max out the efficiency of grope and also energy because interactions (small talk/compliment/flirt) do not seem to depend on any of your skills. If you have nothing to do (unlikely because you can always order supplies or produce serum by yourself since you have no employees early) go to the gym to max out stamina.
Maybe none of this is intended interaction (anyway clearly not optimal), so we can see the above as bugs.
Currently, day 150+, around 8-9 employees. Recently got to 10+ to try to trigger quests (triggered on a clone though, weird interaction...).