A geometry problem by Trường Tùng Nguyễn

A 3 × 3 × 3 cube is made of 1 × 1 × 1 cubes glued together. What is the maximal number of small cubes one can remove so the remaining solid has the following features:

1) Projection of this solid on each face of the original cube is a 3 × 3 square;

2) The resulting solid remains face-connected (from each small cube one can reach any other small cube along a chain of consecutive cubes with common faces).