For Split Payments, we don't change the product inventory until the last (final) payment is made. We have done this to avoid a situation where a product will not be available for future purchases.
Let's assume that you have a product with an initial inventory of 2. Your customer chooses a Shared Payment option and shares it between 3 recipients. The first (initial) customer ordered the product and paid for his part, but we keep inventory the same: 2. The second customer pays for his part of the purchase, but we still keep product inventory the same: 2. When the third customer pays for his part of the purchase, we will change the inventory to 1.
Thus, our app does not change your store inventory until the last Split Payment is paid.
For Installment payments, inventory is updated after the initial payment.