what exactly are you thinking the investigation would be? if it's just something like examining the body of one of sijhen's victims, i don't think that'd be too big of a deal. the players have an opportunity to do it with caius in book 3 itself, and they can't really get much from it (at least, not from the body itself - they can gather quite a bit from surrounding and prior evidence).
So I started the campaign with book 2. I'm giving the party 5 weeks of downtime before we start book 1. In that time, I'm preparing "something" for each of their characters to do for hopefully no longer then like 20-30 minutes real time. Goal is to have the downtime scenario be interesting, short and hint at things upcoming in the campaign.
Our Vekeshi is attending the cult meeting and will learn of the higher ups in the mystics. He'll meet: The Old Stag again, Asrabey Varal, Valentia Yorke, Erdanen Torrance, and Tyler Starke. He's planning on debriefing them on the bleak gate happenings and I plan on having Ekossigan present to hear of it for the first time (will lead to book 5 madness).
Our Aspiring politician Medium will be invited to a high level political gathering with Nigel Price-Hill who will offer a bigger role in the political workings of the city. He'll mention a "job" upcoming that he thinks the party will be well suited for (Its the security detail of book 1). He'll get a chance to meet; Ethelyn of Shale (remember, this is before the events of book 1 but after the events of book 2); Catherine Romana; Erdanen Torrance; and whoever random political figures I find. the gathering will be secretly about politicians swaying eachother to vote for the railroad vs wunderbahn construction.
Our Anti-Gale technologist will meet with Benedict Pemberton. Not exactly sure what this will look like, but will likely have the PC working with Pemberton to create the B.E.A.R for the expo in book 3. Would like for him to meet harkover but not sure how to do that yet.
Our Investigator who is the niece of Margaret Saxby, I have planned to start an investigation into potential disloyal/corrupt constables of the HRC. Perhaps Nigel will give a letter to the medium to secretly deliver to the investigator. My initial plan is to have someone give her a case (Sijen) where evidence has been mysteriously gone missing, witnesses have shut their mouths, the media has been halted access, and general progress on the investigation has been halted. Hopefully, during the investigation she'll draft up a short list of possible corrupt constables. If i'm really lucky, one of the people on the list will be Margaret Saxby.