Why does the Duchess want to sell the pearls in "The Duchess and the Jeweller"?

Virginia Woolf's short story "The Duchess and the Jeweller" focuses on a wealthy London jeweler and his purchase of ten false pearls from the Duchess of Lambourne. The Duchess is described as "very fat" and "past her prime." We also know that she has pawned most of her jewels already, because she emphasizes that these are the last items she can sell.
The Jeweller and the Duchess have an odd relationship:

They were friends, yet enemies; he was master, she was mistress; each cheated the other, each needed the other, each feared the other, each felt this and knew this every time they touched hands thus in the little back room.

This proves that the Duchess has been a regular customer of the Jeweller but that their relationship is fraught with mistrust.
The root of this awkwardness is partially revealed when the Duchess shows the pearls to the Jeweller. His first thought is:

Was she lying again? Did she dare?

This is an important detail. It reveals that the Duchess has a history of deception and has presumably sold the Jeweller false gems in the past. When the Duchess reveals that she has been gambling and had a "bit of bad luck," the Jeweller revels in the powerful knowledge he possesses.
However, the Duchess claims that she desperately needs 20,000 pounds for her three girls:

"Araminta, Daphne, Diana," she moaned. "It's for them."

This revelation, genuine or not, proves a quandary for the Jeweller. The Jeweller "loves" Diana. If the Jeweller refuses to purchase the pearls, he might lose a chance at love. But if he does purchase them, he risks spending 20,000 pounds on worthless baubles. At the end of the story, after signing a cheque, the Jeweller learns the pearls are fake.