Sincere wide $\tau$-tilting modules

Enomoto and Sakai introduced wide $\tau$-tilting modules, which are $\tau$-tilting modules over functorially finite wide subcategories. They also proved that wide $\tau$-tilting modules bijection with doubly functorially finite image-cokernel-extension-closed (ICE-closed) subcategories, which extended Adachi-Iyama-Reiten's result. In this paper, we show that this bijection can be restricted to the support sets. As a consequence, we establish bijections between sincere wide $\tau$-tilting modules, sincere ICE-closed subcategories, and sincere epibricks, and then we show that its number is related to the little Schr $\ddot{o}$der number for Nakayama algebras.