I found a much better connection, through Jean Chabert's wife Anne Lehoux (her name is Lehoux or Le Houx, definitely not Leroux). She is a first cousin of Hélie Lehoux (who, by the way, is my direct ancestor).
This may seem surprising as they come from different places, Rouen and Tours, 300 km apart. But they were a family of silk merchants and were used to travel.
There is no immediate proof (such as a marriage record giving the names of Jean's and Anne's parents, and their places of origin). But there are many hints that she is not from Rouen. First, look at the baptisms of her children (referenced in Fichier origine and transcribed by Archange Godbout, pages 41-42, with minor errors).
At the baptism of their son Henry in 1634 the godfather is Henry Lehoux, an official from Paris (I believe he is Anne Lehoux's uncle).
At the baptism of their son Jean in 1638 the godfather is François Chabert, merchant in Tours, and the godmother is Anne Boullé (we will see later that she is Anne Lehoux's mother, wife of Pierre Lehoux, also merchant in Tours).
At the baptism of their daughter Marguerite in 1640 (the king's daughter) the godfather is Louis Lehoux, merchant (he is Anne Lehoux's brother, also from Tours), and the godmother is Margueritte Jacquet, épouse Maultrot (Louis Maultrot, again from Tours).
At the baptism of their son Pierre in 1642 the godfather is Sébastien Chabert (possible brother of Jean Chabert), and the godmother is Madeleine Lehoux (sister of Anne Lehoux). They will marry each other 3 years later in Tours.
Finally, this document names Anne and her siblings as heirs of Pierre Lehoux and Anne Boullay, but it is quite hard to read.
Jean Chabert and Anne Lehoux probably got married in Tours, Saint-Saturnin parish, before 1631 (date of earliest surviving records). Anne was probably baptized in the same place after 1603, and in theory records are available, but they are not indexed before 1626, and I did not have the courage to search for her name in so many hardly legible pages.
There are three couples Chabert-Lehoux: Jean and Anne, Michel and Catherine, Sébastien and Madeleine. It is clear from the document above that Anne, Catherine and Madeleine are sisters. Their husbands could be brothers. The third mariage is available and it names Sébastien's parents, who may also be Jean's parents: Étienne Chabert and Marie Ralleau. They are not from Tours, but from Thouars, 100 km further away. Some more ascendants are known.
Baptisms in Thouars, Saint-Médard parish are available for that period, and there is indeed a Jean Chabert baptised on 21 Apr 1608. However, he is not Sébastien Chabert's brother, but his uncle. More research is needed here.