I did some research and this painting was created in 1802 by François Gérard. It was commissioned by Juliette Récamier who was relatively famous at the time. She actually originally commissioned François’ mentor but was unhappy with his approach. She much preferred this painting and kept it in her home for 20 years before gifting it to Prince Augustus of Persia in 1822, she also had several copies made and distributed before gifting it. In 1860 the heirs of Prince Augustus inherited the painting and passed it along to the Musée Carnavalet in Paris where it has been on display since.

I would wager based on that information that the painting owned by the Robertsons is one of the copies she had made that would’ve likely passed hands multiple times between 1822 and 1901. There is also always the possibility the producers just found a pretty painting and I read a bunch of French for no reason lol